40 dm12 abstracts ip0 dnes knig prize lecture: talktitle tbd...

40 DM12 Abstracts

IP0

Dnes Knig Prize Lecture: Talk Title TBD

Abstract not available at time of publication.

Zeev DvirPrinceton [email protected]

IP1

Cell Complexes in Combinatorics

Cell complexes of various kinds have been invented intopology to help analyze manifolds and other spaces. Byintroducing a combinatorial structure they make algo-rithms for computing topological invariants possible. Sim-plicial complexes are well-known examples. In the otherdirection, several structures studied in combinatorics nat-urally suggest associated cell complexes. Can the link totopology provided by these cell complexes be of use fordealing with purely combinatorial questions, or are theyjust idle curiosities? The answer is definitely ”yes” in thesimplicial case, as testified by several successes of what hascome to be called ”topological combinatorics”. But whatabout more general cell complexes? In the talk this ques-tion will be discussed. Examples, old and new, arising ingraph, group and number theory will be reviewed.

Anders Bjorner

Royal Institute of Technology (KTH)[email protected]

IP2

On Sidorenko’s Conjecture

The Erdos-Simonovits-Sidorenko conjecture is well-knownin combinatorics but it has equivalent formulations in anal-ysis and probability theory. The shortest formulation is anintegral inequality related to Mayer integrals in statisticalmechanics and Feynman integrals in quantum field theory.We present new progress in the area. Part of the talk isbased on joint results with J.L. Xiang Li. In particular wepresent a type of calculus (based on logarithmic functions)which can be used to prove inequalities between subgraphdensities.

Balazs SzegedyUniversity of TorontoDepartment of [email protected]

IP3

Forcing Large Transitive Subtournamets

The Erdos Hajnal Conjecture states roughly that a graphwith some induced subgraph excluded has a large cliqueor a large stable set. A similar statement can be formu-lated for tournaments (a tournament is an orientation of acomplete graph), replacing cliques and stable sets by tran-sitive subtournaments; and the two conjectures turn outto be equivalent. This talk will survey a number of recentresults related to the latter conjecture. In particular, wewill discuss a new infinite class of tournaments excludingwhich forces large transitive subtournaments; to the bestof our knowledge this is the first such class not obtainedby the so-called substitution operation.

Maria ChudnovskyColumbia

[email protected]

IP4

Adding and Counting

In mathematics, the stuff of partitions seems like merechild’s play. The speaker will explain how the sim-ple task of adding and counting has fascinated many ofthe world’s leading mathematicians: Euler, Ramanujan,Hardy, Rademacher, Dyson, to name a few. And as is typ-ical in number theory, many of the most fundamental (andsimple to state) questions have remained open. In 2010,the speaker, with the support of the American Institute forMathematics and the National Science Foundation, assem-bled an international team of researchers to attack someof these problems. Come hear Professor Ono speak abouttheir findings: new theories which solve some of the famousold questions.

Ken OnoEmory UniversityDepartment of Mathematics and Computer [email protected]

IP5

Coloring 3-colorable Graphs; Graph Theory Fi-nally Strikes Back!

We consider the problem of coloring a 3-colorable graph inpolynomial time using as few colors as possible. Startingwith Wigderson in 1982 (O(n1/2) colors), Blum in 1990

came with the first polynomial improvements (O(n3/8)colors). His improvement is based on graph theoreticalapproach. Karger, Motwani, Sudan in 1994 is the firstto use semi-definite programming (SDP) to give improve-ment, and then Karger and Blum in 1997 combines Blum’smethod with the SDP improvement to show thatO(n0.2142)colors suffices. Since then, the only improvements in semi-definite programming have been made (Arora, Chlamtac,and Charikar in 2006 (O(n0.2111) colors), and Chlamtacin 2007 (O(n0.2072) colors)). We present the first im-provement on the graph theory side since Blum in 1990.With a purely graph theoretical approach, we get down toO(n4/11) colors (over Blum’s O(n3/8) colors). Combiningit with SDP, we get down to O(n0.2038) colors. Joint workwith Mikkel Thorup (ATT Research)

Ken-ichi KawarabayashiNational Institute of Informatics, Japank [email protected]

IP6

Talk Title TBA - Singh


Mona SinghPrinceton [email protected]

IP7

The Hub Labeling Algorithm

This is a survey of Hub Labeling results for general androad networks. Given a weighted graph, a distance oracletakes as an input a pair of vertices and returns the dis-tance between them. The labeling approach to distance

DM12 Abstracts 41

oracle design is to precompute a label for every vertex sothat distances can be computed from the correspondinglabels. This approach has been introduced by [Gavoille etal. ’01], who also introduced the Hub Labeling algorithm(HL). HL has been further studied by [Cohen et al. ’02].We study HL in the context of graphs with small high-way dimension (e.g., road networks). We show that underthis assumption HL labels are small and the queries aresublinear. We also give an approximation algorithm forcomputing small HL labels that uses the fact that short-est path set systems have small VC-dimension. Althoughpolynomial-time, precomputation given by theory is tooslow for continental-size road networks. However, heuris-tics guided by the theory are fast, and compute very smalllabels. This leads to the fastest currently known practi-cal distance oracles for road networks. The simplicity ofHL queries allows their implementation inside of a rela-tional database (e.g., in SQL), and query efficiency assuresreal-time response. Furthermore, including HL data in thedatabase allows efficient implementation of more sophis-ticated location-based queries. This approach brings thepower of location-based services to SQL programmers

Andrew GoldbergMicrosoft Research Silicon [email protected]

IP8

Algorithms, Graph Theory, and the Solution ofLaplacian Linear Equations

We survey several fascinating concepts and algorithms ingraph theory that arise in the design of fast algorithmsfor solving linear equations in the Laplacian matrices ofgraphs. We will begin by explaning why linear equationsin these matrices are so interesting. The problem of solvinglinear equations in these matrices motivates a new notionof what it means for one graph to approximate another.This leads to a problem of graph sparsification–the ap-proximation of a graph by a sparser graph. Our algorithmsfor solving Laplacian linear equations will exploit surpris-ingly strong approximations of graphs by sparse graphs,and even by trees. We will survey the roles that spectralgraph theory, random matrix theory, graph sparsification,low-stretch spanning trees and local clustering algorithmsplay in the design of fast algorithms for solving Laplacianlinear equations.

Daniel SpielmanYale [email protected]

CP1

On Antimagic Vertex Labeling Of Hypergraphs

Let H = (V,E) be a graph with vertex set V (H) and edgeset E(H). Moreover suppose that v = |V (H)|, e = |E(H)|and N denotes the set of non negative integers. An an-timagic (magic) vertex labeling is a bijection “f” fromV (H) to the set of consecutive integers 1, 2, . . . , v if theinduced edge labeling “g” from E(H) to the set of non neg-ative integers N defined by g(e)=? f(v) for all v in E(G) isinjective (constant) function. A hypergraph H is called an-timagic (magic) iff there exist an antimagic (magic) vertexlabeling of H . In this paper we formulate antimagic vertexlabeling on star and disjoint union of star hypergraph.

Akhlaq BhattiFAST-NUCES LAHORE, PAKISTAN

[email protected]

Muhammad JavaidFAST-National University of Computer and EmergingSciencesLahore Campus, Lahore, [email protected]

CP1

Distingushing with Nordhaus Gaddum graphs

Albertson and Collins introduced the distinguishing num-ber in 1996 and Collins and Trenk introduced the distin-guishing chromatic number of a graph which requires thatthe coloring be proper as well as distinguishing in 2006.We revisit the Nordhaus-Gaddum inequalities which givebounds on the sum and the product of χ(G) and χ(G) forany graph G, and give analogues of these inequalities forthe distinguishing chromatic number. We provide a newcharacterization of those graphs that achieve equality forthe upper bound in the sum Nordhaus-Gaddum inequality,which leads to a polynomial-time recognition algorithm forthis class and efficient computation of their chromatic num-bers.

Karen CollinsWesleyan [email protected]

Ann N. TrenkWellesley [email protected]

CP1

On Super (a,d)-Edge Antimagic Total Labeling ofSubdivided Caterpillar

Let G = (V,E) be a graph with v = |V (G)| vertices ande = |E(G)| edges. An(a, d)-edge antimagic total labelingis a bijection “f” from V (G)UE(G)to the set of consec-utive integers 1, 2, .., v + e, such that the weights of theedge W = {w(xy) : xy in E(G)} form an arithmetic pro-gression with the initial term “a” and common difference“d” where w(xy) = f(x) + f(y) + f(xy). “W” is calledthe set of edge-weights of the graph G. Additionally, iff(V ) = 1, 2, ..., |V (G)| then f is called super (a, d)-edgeantimagic total labeling and G is called super(a, d)-edgeantimagic total. In this paper we formulate, super (a, d)-edge antimagic total labeling on subdivided caterpillar fordifferent integral values of d.

Muhammad JavaidFAST-National University of Computer and EmergingSciencesLahore Campus, Lahore, [email protected]

Akhlaq BhattiFAST-NUCES LAHORE, [email protected]

CP1

On Two Conjectures About Graceful Digraphs

A digraph D with p vertices and q arcs is labeled by assign-ing a distinct integer value g(v) from {0, 1, . . . , q} to eachvertex v. The vertex values, in turn, induce a value g(u, v)on each arc (u, v) where g(u, v) = (g(v)−g(u))(mod q+1).

42 DM12 Abstracts

If the arc values are all distinct then the labeling is calleda graceful labeling of digraph. In this talk we present theproofs of two conjectures namely, Bloom and Hsu’s conjec-ture (1985) on graceful unicyclic wheels and Du and Sun’s

conjecture (1994) on the gracefulness of the digraph n.−→Cm.

Shivaraj KumarNITK Surathkalshivaraj [email protected]

Suresh HegdeNITK Surathkal, [email protected]

CP2

Saving Sets of Vertices in the Firefighter Problem

The Firefighter Problem is a discrete-time model of thespread of a fire using a simple graph. At each time step aplayer has an opportunity to defend some vertex (or set ofvertices) and then the fire spreads from all burning verticesto all of their undefended neighbours. Here we examine thedecision problem that asks if a specified set of vertices canbe saved from burning.

Christopher Duffy, Gary MacGillivraryUniversity of [email protected], [email protected]

CP2

A Game on Zero Forcing Sets

We consider the two player game on a graph where playersalternate turns, with one player selecting a vertex to colorblack on each turn and the other player selecting a vertexto color white on each turn. The black player wins if afterall of the vertices have been colored, the black vertices forma zero forcing set. This game will be discussed for severalclasses of graphs.

Kara Greenfield, Alyssa GottshallWorcester Polytechnic [email protected], [email protected]

Steve ButlerUC Los [email protected]

Young MichaelIowa State [email protected]

CP2

A Generalization of the Nim and Wythoff Games

We define a two-player game G(‖, \) (with k ≤ n) as fol-lows: each player chooses up to k stacks from a total ofn stacks and removes an equal number of tokens from theselected stacks. The first player who cannot move loses.G(∞, \) is ordinary Nim and G(∈,∈) is Wythoff. We willpresent a conjecture on the structure of the losing posi-tions of this class of games and prove a special case of theconjecture.

Silvia HeubachCalifornia State University Los Angeles

[email protected]

Matthieu DufourUniversity of Quebec at [email protected]

CP2

Structure of Weighted Graphs with Forbidden Sub-division and Graph Sharing Games

We show that every weighted graph excluding subdivisionof a fixed (small) graph contains one of the following struc-tures:

• a connected subgraph separating the graph into heavybalanced components,

• a heavy set of vertices connected by paths as in acycle.

This characterization yields a strategy for the first player togather a positive fraction of the total weight in a variantof graph sharing game played on graphs with forbiddensubdivision.

Piotr MicekJagellionian [email protected]

Adam GagolMaria Curie-Sklodowska [email protected]

Bartosz WalczakJagiellonian [email protected]

CP3

2-Switches and Isomorphism Classes

A 2-switch is an operation that changes adjacencies in agraph while preserving vertex degrees. The resulting graphmay or may not have the same isomorphism class; we showthat if a 2-switch changes the isomorphism class of a graph,then it takes place in one of four configurations. We alsopresent a sufficient condition for 2-switches to change graphisomorphism classes and examine consequences for graphsdetermined up to isomorphism by their degree sequences.

Michael BarrusBlack Hills State UniversityDepartment of [email protected]

CP3

The Number of Spanning Trees in Self-SimilarGraphs.

The number of spanning trees of a graph is an importantinvariant related to its topological and dynamic proper-ties (reliability, communication, synchronization,etc.) Itscalculation remains a challenge, particularly for large net-works. We describe a method to find an exact analyticalexpression for the number of spanning trees (and tree en-tropy) for several relevant graph families, including Hanoigraphs, Apollonian networks and some maximal outerpla-nar graphs. The method is based on the self-similarity ofthe graphs considered.

Francesc Comellas

DM12 Abstracts 43

Universitat Politecnica de [email protected]

Zhongzhi ZhangFudan University, [email protected]

CP3

On the 2-Edge Clique Cover Numbers of Graphs

Let F be a multifamily of subsets of the vertex set of agraph G. The multifamily F is called a 2-edge clique coverof G if for any two distinct I , J ∈ F , I ∩J is a clique of G,and the family {I ∩ J | I, J ∈ F , I �= J} is an edge cliquecover of G. The minimum size of a 2-edge clique cover of Gis called the 2-edge clique cover number of G. In general,it is hard to compute a 2-edge clique cover of a graph. Inthis talk, we study relationships between the 2-edge cliquecover number and the edge clique cover number of a graph.Then we focus on the 2-edge clique cover numbers of a pathand a cycle, and then give a bound for 2-edge clique covernumber of a star K1,n by using a block design.

Jung Yeun LeeNational Institute of Mathematical ScienceSouth [email protected]

Suh-Ryung KimSeoul National [email protected]

Boram ParkDIMACSRutgers [email protected]

Yoshio SanoNational Institute of [email protected]

CP3

Dual-Chordal and Strongly Dual-Chordal Graphs

If a subgraph is defined to be strength-k whenever it is in ≥k maxcliques, then a graph is (strongly) chordal if and onlyif (for all k ≥ 1) every cycle of (strength-k) edges eitherhas a (strength-k) chord or is a (strength-k) triangle. Acarefully crafted notion of dual strength (using cycle/cutsetduality) allows parallel characterizations of (strongly) dual-chordal graphs and a listing of all 3-connected stronglydual-chordal graphs.

Terry McKeeWright State UniversityDepartment of Mathematics & [email protected]

CP3

Well-Covered Graphs Without C4, C5, C6

Let G = (V,E) be a graph and w : V −→ R be a weightfunction. Then G is w-well-covered if all its maximal in-dependent sets are of the same weight. The set of weightfunctions w such that G is w-well-covered is a vector space.Given an input graph G without cycles of length 4, 5, and6 (not necessarily induced), we characterize polynomiallythe vector space of weight functions w for which G is w-

well-covered.

David TankusAriel University Center of Samaria, [email protected]

Vadim E. LevitAriel University Center of Samaria and Holon Inst. ofTech.Dept. of Computer Science & [email protected]

CP4

Construction of nonuniform p-wavelet packets

A generalization of Mallat’s classical multiresolution anal-ysis for which the translation set is a discrete set which isnot a group, was defined by Gabardo and Nashed. In thistalk, we introduce nonuniform multiresolution p-analysison L2(R+), analogy of Gabardo-Nashed definition and thenwe will construct the associated p-wavelet packets for suchan multiresolution p-analysis on L2(R+).

Jahangir Cheshmavar

Payame Noor University (PNU)[email protected]

CP4

Application of New Variational Method UsingHamiltonian for Nonlinear Oscillators with Discon-tinuities

In this paper, we used Hamiltonian for nonlinear oscillatorswith discontinuities. The maximal relative error for thefrequency obtained by new variational method comparedwith the exact solution indicates the remarkable precisionof this method. Some examples are given to illustrate theeffectiveness and convenience of the method

Waseem A. KhanCIIT, ISLAMABAD, [email protected]

CP4

A Comparison of Conventional and Fuzzy Methodsin Performance Appraisal Process

The purpose of this paper is to address the performance ap-praisal process via conventional and Fuzzy satisfied meth-ods.Conventional methods for solving performance ap-praisal decision making problem are by using statisticalmethods(chi-square, correlation etc)on the measurement ofperformance based on specific appraisal criteria.This re-search shows that the Fuzzy satisfied method is an easy tounderstand and realistic technique as it enables us to un-derstand the relative strengths and weaknesses of employ-ees,thus providing useful information for improving com-petitive advantage.

Idowu A. OsinugaDepartment of Mathematics, College of Natural Sciences,Federal University of Agriculture, Abeokuta, [email protected]

Adegboyega AdekoyaDepartment of Mathematical Sciences, Olabisi OnabanjoUnivAgo-Iwoye, Nigeria

44 DM12 Abstracts

[email protected]

CP4

On the Structure of Some Groups ContainingL2(13)wrl2(17)

In this paper, we will show the structure of some groupscontaining the wreath product L2(13)wrL2(17).Some sym-metric and alternating groups are constructed in view ofsuch wreath product. Some other related cases are alsoincluded.

Basmah ShafeeUmm Al qura Universitydr.basmah [email protected]

CP4

Fixed Points Of a Fuzzy Semantics Function

The meaning of a program is given by specifying the func-tion (from input to output) that corresponds to the pro-gram. The denotational semantic definition, thus mapssyntactical things into functions. A fuzzy relational se-mantics is a mapping of programs to relations. We considerthat the input-output semantics of a program is given bya fuzzy relation on its set of states. In a nondeterministiccontext, this relation is calculated by considering the worstbehavior of the program (demonic fuzzy relational seman-tics). In this paper, we concentrate on while loops. We willpresent some interesting results about the fixed points ofthe while fuzzy semantics function; f(X) = Q∪f uzPfuzXwhere PQ =, by taking P := tB and Q := t, one gets thedemonic semantics we have assigned to while loops in pre-vious papers. We will show that the least angelic fixedpoint is equal to the greatest demonic fixed point of thefuzzy semantics function.

Fairouz TchierKing saud [email protected]

CP4

Binary Sequences, Fractals, and Primes

The concept of a fractal transformation is discussed. A1-dimensional example leads to connections between thetopics in the title.

Andrew VinceDepartment of MathematicsUniversity of [email protected]

Michael BarnsleyAustralian National [email protected]

CP5

Order Dimension and Coloring of Planar Point Sets

The Erdos-Szekeres Conjecture of planar point sets in gen-eral position will be discussed using convex geometries oranti-matroids. We study this problem with two graphs,one based on the lattice of closed sets and the other fromthe copoints of the convex geometry. We use these graphsto show that any point set in general position with at least

2n−2 + 1 has order dimension at least n.

Jonathan E. BeagleyGeorge Mason [email protected]

CP5

Universal Cycles for Weak Orders and a Problemof Knuth

A weak order on [n] = {1, 2, ..., n} is a string w1w2...wn

such that {w1, w2, ..., wn} = [h], for some height h ≤ n.Knuth asked: For which n does there exist a Gray code forall weak orders on [n] using operations of the form wi ↔wi+1 and wi ← wi+1? We prove such existence for a class ofweak orders with the additional operation wi ↔ wi+2 anddiscuss open problems for fixed height weak orders that arerelated to solving the original problem. Furthermore, wepresent constructions of universal cycles for weak orders,as well as for weak orders of fixed height or weight.

Victoria Horan, Glenn HurlbertArizona State [email protected], [email protected]

CP5

Zeta Polynomials for Shellable Posets and TheirApplications

The zeta polynomial Z(P, n) for a poset P is the number ofmultichains in P of length n−2, and a shellable poset P isa graded poset whose maximal chains explain the structureof chains in P . Using shellability, we establish theorems oncomputations of zeta polynomials for shellable posets. Ap-plying these theorem, we also derive new resluts on multi-dimensional partitions, multidimensional Young tableaux,lattice paths, and multiset permutations.

Joon Yop LeeASARC, [email protected]

CP5

The Largest Size Versus the Largest Weight ofFamilies of Sets Without a Poset P

We are interested in how large a family of subsets of[n] := {1, 2 . . . , n} not containing a given poset P as asubposet can be. Such a family is called a P -free fam-ily and the largest size of a P -free family is denotedby La(n, P ). Griggs and Lu conjectured that π(P ) :=

limn→∞ La(n, P )/(

n�n

2�)

exists and is an integer for any fi-

nite P . The Lubell function of a family F of subsets of

[n] is hn(F) :=∑

E∈F( \|E|)−∞

which can be viewed as the

weighted sum of sets in F . It gives an upper bound of |F|,which is crucial in solving the forbidden subposet problemabove. We will study the limit of λn(P ) := maxF hn(F)over all P -free families F as n increases and demonstrate aclass of posets satisfying π(P ) = limn λn(P ). We will alsoshow that for many posets limn λn(P ) exists and can beevaluated, although π(P ) is unknown.

Jerry GriggsUniversity of South Carolina, [email protected]

Wei-Tian LiDepartment of Mathematics

DM12 Abstracts 45

Univeristy of South [email protected]

CP5

Searching Algorithms in Partially Ordered Set

Using conceptual graphs and graph homomorphism it ispossible to build a basic query-answering mechanism basedon semantic search. Because graph homomorphism isNP-Complete, the main requirement for conceptual graphdatabase building and managing algorithms is to reducethe homomorphism checks. Graph homomorphism definesa partial order over conceptual graphs. Searching is a basicoperation for building and managing operations as order-ing, updating, and retrieval. We propose two new searchingalgorithms in a poset.

Abdurashid R. MamadolimovMIMOS, [email protected]

CP5

The Bkr Inequality on Finite Distributive Lattices

The van den Berg-Kesten-Reimer inequality, or BKR in-equality, is a well-known result in combinatorial probabil-ity with applications in percolation. The BKR inequalityapplies to product spaces X equipped with product prob-ability measures. For events A and B, the box product ofA and B is the set of tuples x in X that lie in A and Bdisjointly, i.e. x lies in A due to the values of one set of x’scoordinates and in B due to a disjoint set of coordinates.The BKR inequality states that for any two events A and Bof any finite product probability space the measure of thebox product of A and B is less than or equal to the productof the measures of A and B. We extend the box productand the BKR inequality to finite distributive lattices withlog modular measures, analogues of product measures. Aproduct space can be made into a distributive lattice by lin-early ordering each member of the product. The new boxproduct of A and B will then contain the old box product.Furthermore the containment is often strict, improving theBKR inequality.

Clifford D. SmythUniversity or North Carolina [email protected]

CP6

On the Locating-Chromatic Number of a CoronaProduct of Two Graphs

The problem of determining the locating-chromatic num-ber for a general graph is an NP-hard problem. Therefore,there is no efficient algorithm to determine this numberof arbitrary graph. Some heuristics have been developedand some studies were done to find the locating-chromaticnumber for certain graphs. In this paper, we investigatethe locating-chromatic number for a corona product G H .Some upper bound will be derived. We also determine thelocating-chromatic number of certain graphs.

Edy T. Baskoro, Ira PurwasihInstitut Teknologi [email protected], ira apni [email protected]

CP6

On The Structure Of The γ-Graph Of A Tree

The γ-graph of a graph G has vertex set comprised of allminimum dominating sets of G. There are two ways todefine adjacency in the γ-graph: where dominating setsare adjacent if they agree in all but one vertex, or wheredominating sets are adjacent if they agree in all but onevertex and the differing vertices are adjacent. Here weanswer open questions on maximum degree, diameter, andorder of γ-graphs for trees.

Michelle Edwards, Gary MacgillivrayMathematics and StatisticsUniversity of [email protected], [email protected]

CP6

Some Unsolved Problems on Cycles

Hajos’ conjecture that every simple even graph on n ver-tices can be decomposed into at most (n − 1)/2 cycles.Let f(n) be the maximum number of edges in a graph onn vertices in which no two cycles have the same length.P. Erdos raised the problem of determining f(n). P. Er-dos conjectured that there exists a positive constant c suchthat ex(n,C2k) ≥ cn1+1/k. This talk summarizes some re-sults on these problems and the conjectures that relate tothese. It seems to me that Hajos conjecture is false.

Chunhui LaiZhangzhou Normal [email protected]

Mingjing LiuDepartment of Mathematics and Information ScienceZhangzhou Normal [email protected]

CP6

On Signed Star Domination and Domatic Numbersof Complete Multipartite Graphs

The signed star domination and domatic numbers of agraph G, γSS(G) and dSS(G), were first introduced in[B. Xu, On edge domination numbers of graphs, DiscreteMathematics, 294 (2005), 311–316.] and [M. Atapour,S. M. Sheikholeslami, A. N. Ghameshlou and L. Volk-mann, Signed star domatic number of a graph, DiscreteApplied Mathematics, 158 (2010), 213–218.]. We give newconstructions to complete the proofs of γSS(Km,n) anddSS(Km,n). We also obtain γSS(Km(n)) and dSS(Km(n)).

Wu-Hsiung LinDepartement of Applied MathematicsNational Chiao Tung [email protected]

Chiuyuan Chen, Well Y. Chiu, ALLEN Y. WangDepartement of Applied Mathematics,National Chiao Tung [email protected], [email protected], [email protected]

CP6

Decompositions of Complete Multipartite Graphs

46 DM12 Abstracts

into Gregarious Long Cycles

The notion of gregarious cycles in complete multipartitegraphs was introduced by Billington and Hoffman in 2003and was modified by Billington, Smith, and Hoffman in2007. In this talk, we propose more general definition of“gregarious” cycles which is a common generalization ofboth the definitions. With our definition, we can considerthe problem to characterize complete multipartite graphswhich are decomposable into gregarious long cycles, andwe give some results on this problem.

Jeongmi Park, Jung Rae ChoDepartment of MathematicsPusan National [email protected], [email protected]


CP6

A New Upper Bound for the Broadcast Domina-tion Number of a Graph

The broadcast domination number of a graph G, de-noted γb(G), is the minimum cost of a dominatingbroadcast on G. We present an improvement to thewell-known Bollobas-Cockayne bound from γb(G) ≤min {rad(G), γ(G)} ≤ 2 ir(G)− 1, to γb(G) ≤ 3

2ir(G). We

also show the existence of a graphG with γb(G)− ir(G) = kfor any positive integer k.

Laura E. Teshima, Christina MynhardtUniversity of [email protected], c.m. mynhardt

CP7

The Edge Density of Critical Digraphs

Let χ(G) denote the chromatic number of a graph G. Wesay that G is k-critical if χ(G) = k and χ(H) < k for everyproper subgraph H ⊂ G. Over the years, many prop-erties of k-critical graphs have been discovered, includingimproved upper and lower bounds for ||G||, the number ofedges in a k-critical graph, as a function of |G|, the numberof vertices. In this note, we analyze this edge density prob-lem for directed graphs, where the chromatic number χ(D)of a digraph D is defined to be the fewest number of coloursneeded to colour the vertices of D so that each colour classinduces an acyclic subgraph. For each k ≥ 3, we con-struct an infinite family of sparse k-critical digraphs for

which ||D|| < ( k2−k+12

)|D| and an infinite family of dense

k-critical digraphs for which ||D|| >(12− 1

2k−1

)|D|2. One

corollary of our results is an explicit construction of an in-finite family of k-critical digraphs of digirth l, for any pairof integers k, l ≥ 3. This extends a result by Bokal et.al. who used a probabilistic approach to demonstrate theexistence of one such digraph.

Richard Hoshino, Ken-Ichi KawarabayashiNational Institute of [email protected], k [email protected]

CP7

On the Density of Nearly Regular Graphs with a

Good Edge-Labelling

A good edge-labelling of a simple graph is a labelling of itsedges with real numbers such that, for any ordered pair ofvertices (u, v), there is at most one non-decreasing pathfrom u to v. Say a graph is good if it admits a goodedge-labelling, and is bad otherwise. Our main result isthat any good n-vertex graph whose maximum degree iswithin a constant factor of its average degree (in particu-

lar, any good regular graph) has at most n1+o(1) edges. Asa corollary, we show that there are bad graphs with arbi-trarily large girth, answering a question of Bode, Farzadand Theis. We also prove that for any Δ, there is a g suchthat any graph with maximum degree at most Δ and girthat least g is good.

Abbas MehrabianDepartment of Combinatorics and OptimizationUniversity of [email protected]

CP7

Bounds on Shannon Capacity and Ramsey Num-bers from Product of Graphs

We study Shannon capacity of channels in the context ofRamsey numbers. We overview some of the results on chan-nel capacity, and how Ramsey-type constructions may en-hance them. A new lower bound for a special type of multi-color Ramsey numbers is presented, which implies that thesupremum of the Shannon capacity over all graphs with in-dependence 2 cannot be achieved by any finite graph power.This generalizes to graphs with any bounded independencenumber.

Xiaodong XuGuangxi Academy of [email protected]

Stanis�law P. RadziszowskiDepartment of Computer ScienceRochester Institute of [email protected]

CP7

Colorings of Uniform Hypergraphs with LargeGirth

The talk deals with a combinatorial problem concerningcolorings of uniform hypergraphs with large girth. Let Hbe a hypergraph and Δ(H) denote the maximum vertexdegree ofH . By using a continuous-time random recoloringprocess we prove that if H is an n-uniform non-r-colorablehypergraph with girth at least 6 then

Δ(H) ≥ c rn−1

lnn,

where c > 0 is an absolute constant.

Dmitry A. ShabanovMoscow State [email protected]

Andrei KapavskiiMoscow Institute of Physics and [email protected]

DM12 Abstracts 47

CP7

Embedding Spanning Bipartite Graphs of SmallBandwidth

Bottcher, Schacht and Taraz gave a condition on the mini-mum degree of a graph G on n vertices that ensures G con-tains every r-chromatic graph H on n vertices of boundeddegree and of bandwidth o(n), thereby proving a conjec-ture of Bollobas and Komlos. We strengthen this result inthe case when H is bipartite by relaxing the condition onG to a condition on the degree sequence of G.

Andrew Treglown, Fiachra KnoxUniversity of [email protected], [email protected]

CP8

Forbidden Submatrices

We consider an extremal hypergraph problem, where theorder of the elements and of the hyperedges matter. Inmatrix terms, let F be a given (0,1)-matrix. Among all m-rowed (0,1)-matrices with no repeated columns and no sub-matrix F , what is the maximum number of columns? Wecall this fs(m,F ). It was conjectured by Anstee, Frankl,Furedi, Pach that if F has k rows then fs(m,F ) is O(mk).We give results for 2-rowed F .

Richard P. AnsteeMathematics DepartmentUniversity of British [email protected]

Ruiyuan ChenUBCVancouver, [email protected]

Attila SaliRenyi [email protected]

CP8

Intertwining Connectivities in Matroids Repre-sentable over a Finite Field

Let N1 and N2 be matroids. An intertwine of N1 and N2

is a matroid that has both an N1- and N2-minor, and isminor-minimal with this property. Bonin proved that thereexist matroids N1 and N2 with infinitely many intertwines.We prove that if we instead intertwine “connectivities” in-stead of minors, then are a finite number of F-representableintertwines, for F finite.

Tony HuynhUniversity of [email protected]

Bert GerardsCWI [email protected]

Stefan van ZwamPrinceton [email protected]

CP8

The Minimum Rank of Universal Adjacency Ma-trices

For a simple graph G on n vertices, a matrix of the form

U(α, β, γ, δ) = αA+ βI + γJ + δD

where A is the 0,1-adjacency matrix of G, J is the all onesmatrix of size n, I is the identity matrix of size n, and Dis the diagonal matrix with the degrees of the vertices inthe main diagonal, and α �= 0, β, γ, δ are scalars, is called auniversal adjacency matrix of G. An analogous parameterto the minimum rank of a given graph G is the minimumrank over all matrices in the set of universal adjacencymatrices of G. This parameter is called the minimum uni-versal rank of G, and is denoted by mur(G). Graphs withmur(G) equal to zero and one are characterized. The min-imum universal rank of some families of graphs such ascomplete graphs, complete bipartite graphs, paths and cy-cles are presented. A formula for the minimum universalrank of a regular graph is given. Finally, it is shown thatmur(G) is not monotone on induced subgraphs.This is joint work with Discrete Mathematics ResearchGroup at the University of Regina.

Shahla Nasserasr, Bahman Ahmadi, FatemehAlinaghipourUniversity of [email protected], [email protected],[email protected]

Shaun M. FallatUniversity of [email protected]

Yi-Zheng FanAnhui [email protected]

Karen MeagherUniversity of ReginaSaskatchewan, [email protected]

CP8

The Supertail of a Subspace Partition

Let V = V (n, q) be a vector space of dimension n over thefinite field with q elements, and let d1 < d2 < . . . < dm bethe dimensions that occur in a subspace partition P of V .Let σq(n, t) denote the minimum size of a subspace parti-tion of V in which t is the largest dimension of a subspace.For any integer s, with 1 < s ≤ m, the set of subspaces inP of dimension less than ds is called the s-supertail of P .We show that the number of subspaces in an s-supertail isat least σq(ds, ds−1).

Esmeralda L. NastaseXavier [email protected]

Olof [email protected]

Julianne LehmannBremen [email protected] ¡jule.germany@googlema

48 DM12 Abstracts

Papa SissokhoIllinois State [email protected]

CP8

Rank-Width and Well-Quasi-Ordering of Skew-Symmetric Or Symmetric Matrices

We prove that every infinite sequence of skew-symmetricor symmetric matrices M1, M2, . . . over a fixed finite fieldmust have a pair Mi, Mj (i < j) such that Mi is iso-morphic to a principal submatrix of the Schur complementof a nonsingular principal submatrix in Mj , if those ma-trices have bounded rank-width. This generalizes threetheorems on well-quasi-ordering of graphs or matroids ad-mitting good tree-like decompositions; (1) Robertson andSeymour’s theorem for graphs of bounded tree-width, (2)Geelen, Gerards, and Whittle’s theorem for matroids rep-resentable over a fixed finite field having bounded branch-width, and (3) Oum’s theorem for graphs of bounded rank-width with respect to pivot-minors.

Sang-Il [email protected]

CP9

Applications of the Traveling Salesman Problemand Perfect B-Matching for Finding Genomic Me-dians

Genomic medians model an ancestor of several species.There has been extensive research on finding medians. Westudy the Breakpoint Median Problem (BMP) and presenta novel and successful approach in which the NP-hard prob-lem of BMP for linear multichromosomal genomes is trans-formed into a variation of the Traveling Salesman Problem.We also use prefect b-matchings to show, for the first time,that BMP for circular or mixed unsigned genomes is solv-able in polynomial time.

Maryam Haghighi, Sylvia BoydUniversity of [email protected], [email protected]

CP9

A Branch, Bound, and Remember Algorithm forthe Simple Assembly Line Balancing Problem

A new exact algorithm is presented for the assembly linebalancing problem, which finds and verifies the optimalsolution for every problem in the benchmarks of Hoffmann,Talbot, and Scholl one-half second per problem, on average,including one problem that has remained open for over tenyears. The previous best known algorithm solved 257 of the269 benchmarks. The new algorithm is based on a branchand bound method using memory to eliminate redundantsubproblems.

Sheldon H. JacobsonUniversity of IllinoisDept of Computer [email protected]

Edward SewellSouthern Illinois University [email protected]

CP9

An Approximation Algorithm for the MultilevelBottleneck Assignment Problem

We deal with the multilevel bottleneck assignment prob-lem. The multilevel bottleneck assignment problem is NP-complete, and some approximation algorithms have beenproposed. We propose an approximation algorithm thatuses the element’s rank in each set sorted in the ascend-ing order. We show the effectiveness of our algorithm bynumerical experiments for the case that elements’ valuesfollow the normal distribution. Also we present the idea ofour algorithm’s extension to multidimensional case.

Yuusaku KamuraTokyo A&T [email protected]

Mario NakamoriTokyo University of Agriculture and [email protected]

CP9

Greedy Is Good to Approximate Minimum Rain-bow Subgraphs

We consider the Minimum Rainbow Subgraph problem(MRS): Given a graph G, whose edges are coloured withp colours. Find a subgraph F ⊆ G of minimum order andwith p edges such that each colour occurs exactly once. Inthis talk we will show that the Greedy algorithm for theMRS problem has an approximation ratio of Δ

2+ lnΔ+1

2for graphs with maximum degree Δ. If the average degreed of a minimum rainbow subgraph is known, then the ap-

proximation ratio is d2

+ ln�d�+12

.

Ingo SchiermeyerTechnische Universitaet Bergakademie FreibergInstitut fuer Diskrete Mathematik und [email protected]

CP10

Constructing Self-Dual Graph Embeddings in Sur-faces

A cellular automorphism of a surface S is a 2-cell graphembedding G in S together with an automorphism of thecellular structure given by G. We detail methods whichallow one, given a catalog of all cellular automorphismsin any particular surface S (such as we have produced forfive surfaces), to construct all self-dual graph embeddingsin S. Our methods account for all free and all non-freeautomorphisms, of all possible orders.

Lowell AbramsDepartment of MathematicsThe George Washington [email protected]

Daniel SlilatyDepartment of Mathematics and StatisticsWright State [email protected]

CP10

k-Metric Dimension of Graphs

Let G(V,E) be a connected graph. A subset S of V is said

DM12 Abstracts 49

to be k-resolving set of G, if for every pair of distinct ver-tices u, v ∈ V − S, there exists a vertex w ∈ S such that|d(u,w)− d(v, w)| ≥ k, for some k ∈ Z+. Among all k-resolving sets of G, the set having minimum cardinality iscalled a k-metric basis of G and its cardinality is called thek-metric dimension of G, denoted by βk(G). In this paper,we characterize the graphs with prescribed k-metric dimen-sions and also extend some of the earlier known results onmetric dimension of graphs.

Sooryanarayana BDr. Ambedkar Institute of Technology, Bangaloredr [email protected]

CP10

Almost Empty Monochromatic Triangles in PlanarPoint Sets

For positive integers c, s ≥ 1, let M(c, s) be the least in-teger such that any set of at least M(c, s) points in theplane, no three on a line and colored with c colors, containsa monochromatic triangle with at most s interior points.The case s = 0, which corresponds to empty monochro-matic triangles, has been studied extensively over the lastfew years. In particular, it is known that M(1, 0) = 3,M(2, 0) = 9 and M(c, 0) = ∞, for c ≥ 3. In this paperwe extend these results when c ≥ 4 and s ≥ 1. We provethat the least integer λ(c) such that M(c, λ(c)) <∞ sat-isfies:

�c�p − 1

2≤ λ(c) ≤ c− 2,

where c ≥ 2 and �c�p is the largest prime number lessthan or equal to c. We also show that M(c, c − 2) ≤(c+ 1)2, for c ≥ 2. Finally, we determine the exact valuesof M(c, s) for several small values of c and s, and discussrelated problems for general convex polygons.

Bhaswar B. BhattacharyaDepartment of StatisticsStanford [email protected]

Deepan BasuIndian Statistical Institute, [email protected]

Sandip DasISI, Calcutta, [email protected]

CP10

On The Metric Dimension Of Generalized WheelAnd Other Graphs

For an ordered set W = {w1, w2, ..., wk} of vertices anda vertex v in a connected graph G, the representationof v with respect to W is the ordered k-tuple r(v |W ) = (d(v,w1), d(v, w2), ..., d(v, wk)), where d(x, y) rep-resents the distance between the vertices x and y. The setW is called a resolving set for G if every two vertices of Ghave distinct representations. A resolving set containing aminimum number of vertices is called a basis for G. Theelements of a metric basis are called land marks and themetric dimension of G, denoted by β(G), is the number ofvertices in a basis of G.A generalized wheel graph is Wmn = Cm + Kn. In thisPaper we determine the metric dimensions of GeneralizedWheel graphs. We also obtain the metric dimension of

some cellular network graphs and their dual graphs.

Shreedhar KKVG College of Engineering, [email protected]


CP10

Latin Squares and Competition Numbers

In this talk we show how latin squares are used to findbounds on the competition numbers of complete multipar-tite graphs. Let G be a graph and let Ik denote the graphon k isolated vertices. The minimum k such that G∪ Ik isthe competition graph of an acyclic digraph is called thecompetition number of G. We show that if n ≥ 5 is odd,then k(K4

n) ∈ {n2 − 4n+ 7, n2 − 4n + 8}. For the generalcase, we show that if n is a prime integer and m ≤ n, thenk(Km

n ) ≤ n2 − 2n+ 3.

Jaromy S. KuhlUniversity of West [email protected]

CP10

Metric Dimension of Amalgamation of Graphs

A set of vertices S resolves a graph G if every vertex isuniquely determined by its vector of distances to the ver-tices in S. The metric dimension of G is the minimumcardinality of resolving set of G. Let {G1, G2, . . . , Gn} bea finite collection of graphs and each Gi has a fixed vertexv0i called a terminal. The amalgamation Amal{Gi; v0i} isformed by taking all the Gi’s and identifying their termi-nals. Here we study the metric dimension of Amal{Gi; v0i}for {G1, G2, . . . , Gn} a finite collection of arbitrary graphs.We give lower and upper bounds for the dimension, showthat the bounds are tight, and construct infinitely manygraphs for each possible value of dimensions.

Rinovia Simanjuntak, SALADDIN Uttunggadewa, SuhadiWido SaputroInstitut Teknologi [email protected], s [email protected],[email protected]

CP11

The Periodicity of Winning/Losing States in Sub-traction Games

A subtraction game is a game involving a pile of tokensand a finite set S of positive integers. Two players movealternately, each removing some m tokens provided that mis an element in S. The player who makes the last movewins. We introduce an observation on the periodicity ofthe winning/losing positions of subtraction games whichhave a connection to the periodicity of the nim-sequences.

Nhan Bao HoLa Trobe [email protected]

CP11

Towards a De Bruijn-Erdos Theorem in the L1-

50 DM12 Abstracts

Metric

In 1948, de Bruijn and Erdos proved that every set of npoints in the plane is either collinear, or it induces at leastn lines. The notion of a line can be extended in a nat-ural way to an arbitrary metric space as follows. If δ isa metric, we say that the point x is between the points aand b if δ(a, b) = δ(a, x) + δ(x, b). The line induced by thepoints a and b consists of the points a, b, and all pointsx such that one of a, b, x is between the other two. Withthe usual Euclidean, or L2, metric, this is equivalent to theusual definition of a line. Chen and Chvatal conjecturedin 2006 that a statement analogous to the de Bruijn–Erdostheorem holds in any finite metric space. Despite severalpartial results, this question is still open. We prove thatwith the L1 (also called Manhattan) metric, every set of npoints in the plane is either collinear, or it induces at leastΩ(n6/7−o(1)) lines.

Ida KantorCharles [email protected]

Balazs PatkosAlfr’{e}d R’{e}nyi Institute of Mathematics, [email protected]

CP11

Radio Number of kth Power of a Path

Let G be a connected graph. For any two vertices u and v,let d(u, v) denotes the distance between u and v in G. Themaximum distance between any pair of vertices is calledthe diameter of G and is denoted by diam(G). A radiolabeling of a connected graph G is an assignment of distinctpositive integers to the vertices of G, with x ∈ V (G) labeledf(x), such that |f(u)−f(v)|+d(u, v) ≥ 1+diam(G) for allu, v ∈ V, u �= v. The radio number rn(f) of a radio labelingf of G is the maximum label assigned by f to a vertex ofG. The radio number rn(G) of G is the min{rn(f)} overall radio labeling f of G. The kth power of a graph G,denoted by Gk, is the graph on the vertices of G with twovertices u and v are adjacent in Gk, whenever d(u, v) ≤ kin G. In this paper, we completely determine the radionumber of the graph P k

n for positive integers n, k with k ≥2.

P. Devadas RaoSrinivas Institute of [email protected]


Chandru HegdePES Institute of Technology, [email protected]

CP11

Discrete Operators Applied to Further Generalizethe Integral Image Algorithm

The ”Integral Image” algorithm was extended to continu-ous and more general domains by Wang, Doretto et al in2007. Their theorem will be abbreviated the ”Antideriva-

tive Formula” in this talk. The main results are: (1) a morerigorous formulation (via a discretization of the derivative)to a parameter at the antiderivative formula, and (2) an ex-tension of the antiderivative formula to more general typesof domains, via a (novel) discrete integration method overcurves in the plane.

Amir ShacharMathematical Institute, Hebrew University, Jerusalem,[email protected]

CP11

Graphs with Minimum Identifying Code and Min-imum Size

In a graph G, let B(u) be the set of u with all its neigh-bors in G. A subset S of vertices is called an identifyingcode of G if, for every pair of distinct vertices u and v,both B(u) ∩ S and B(v) ∩ S are nonempty and distinct.A minimum identifying code of a graph G is an identi-fying code of G with minimum cardinality and M(G) isthe cardinality of a minimum identifying code for G. Aminimum identifying code graph G of order n is a graphwith M(G) = �log2(n + 1)� having the minimum numberof edges. Moncel constructed minimum identifying codegraphs of order 2m − 1 for m ≥ 2 and left the same prob-lem, for arbitrary order, to be open. In this talk, we pur-posed the construction of connected minimum identifyingcode graphs to solve this problem.

Li-Da TongNational Sun Yat-sen University, [email protected]

CP12

Eulerian Circuits with No Monochromatic Transi-tions.

Let G be an eulerian digraph with a fixed edge coloring(not necessarily a proper edge coloring). A compatible cir-cuit T is an eulerian circuit of G such that no consecutiveedges in the circuit have the same color. We provide suf-ficient conditions for the existence of a compatible circuit,strengthening results of Fleischner and Isaak. We also showseveral examples of digraphs that do not have compatiblecircuits.

James CarraherUniversity of [email protected]

Stephen HartkeDepartment of MathematicsUniversity of Nebraska - [email protected]

CP12

Labeled Embeddings of Graphs

We present a recent variant of the graph embedding prob-lem on labeled graphs. Given a graph G = (V,E), a k-labeled embedding of G is a labeling f : V → {1..k} suchthat there exists an edge-disjoint placement of two copiesof G into the complete graph K|V | preserving f . The ob-jective consists in maximizing the number of labels k. Wegive upper bounds in the general case, exact values forsome families of graphs, and extensions for the placement

DM12 Abstracts 51

of more than two copies of G.

Eric Duchene, Hamamache KheddouciLyon 1 [email protected],[email protected]

Richard NowakowskiDepartment of Mathematics and StatisticsDalhousie [email protected]

Amine TahraouiLyon 1 [email protected]

CP12

Dynamic Coloring and List Dynamic Coloring ofPlanar Graphs

A dynamic chromatic number χd(G) of a graph G is theleast number k such that G has a proper k-coloring of thevertex set V (G) so that for each vertex of degree at least 2,its neighbors receive at least two distinct colors. We showthat χd(G) ≤ 4 for every planar graph except C5. The listdynamic chromatic number chd(G) of G is the least num-ber k such that for any assignment of k-element lists tothe vertices of G, there is a dynamic coloring of G wherethe color on each vertex is chosen from its list. Based onThomassen’s result that every planar graph is 5-choosable,an interesting question is whether the list dynamic chro-matic number of every planar graph is at most 5 or not.We answer this question by showing that chd(G) ≤ 5 forevery planar graph.

Seog-Jin KimKonkuk [email protected]

Sang June LeeEmory University, [email protected]

Won-Jin ParkSeoul National University, [email protected]

CP12

The Competition Graphs of Powers of Digraphs

The competition graph C(D) of a digraph D has the samevertex set as D and has an edge between vertices u andv if and only if there exists a common prey of u and v inD. Given a digraph D, the convergence of the sequence{C(Dm)}∞m=1 has been studied by several researches thatare related to the notion of competition index. In thispaper, given a digraph D, we focus on the limit graph of thesequence {C(Dm)}∞m=1 if it converges. We find the limitgraph of {C(Dm)}∞m=1 when D has at most two stronglyconnected components, and then we give a sufficient andnecessary condition for its limit graph consisting of onlycomplete connected components.

Boram ParkDIMACSRutgers [email protected]

Woongbae Park, Suh-Ryung KimSeoul National [email protected], [email protected]

CP12

Flow-Continuous Mappings – Influence of theGroup

Mapping f : E(G) → E(H) is M -flow-continuous if forevery flow φ on H using values in M the composition φ ◦ fis a flow on G. This notion was introduced by Jaeger toapproach flow-related conjectures (like CDC). We discussthe influence of the group on flow-continuous mappings.On one hand, groups for which this mapping exists exhibitnice structure. On the other hand, in the important caseof snarks we can restrict to Z2, Z3 or Z.

Robert SamalCharles [email protected]

Jaroslav Nesetrildepartment of Applied MathematicsCharles [email protected]

CP12

Total Weight Choosability of Cartesian Product ofGraphs

A graph G = (V,E) is called (k, k′)-choosable if the follow-ing is true: For any total list assignment L which assignsto each vertex x a set L(x) of k real numbers, and as-signs to each edge e a set L(e) of k′ real numbers, thereis a mapping f : V ∪ E → R such that f(y) ∈ L(y) forany y ∈ V ∪ E and for any two adjacent vertices x, x′,∑

e∈E(x)f(e) + f(x) �= ∑

e∈E(x′) f(e) + f(x′). In this talk,

we will show that if G is the Cartesian product of an evennumber of even cycles, or the Cartesian product of an oddnumber of even cycles and at least one of the cycles haslength 4n for some positive integer n, then G is (1, 3)-choosable.

Jiaojiao Wu, Tsai-Lien WongDepartment of Applied MathematicsNational Sun Yat-sen [email protected], [email protected]

Xuding ZhuZhejiang Normal University, [email protected]

CP13

Counting Minimal Sturmian Words

The subword complexity pw(n) of a word w is the numberof distinct subwords of length n of w. We call a word wSturmian of order N, if pw(n) = n + 1 for n = 1, . . . , N .It is minimal if it has minimal length among all Sturmianwords of order N . Equivalently, it is minimal if it haslength 2N . Recently, Blanchet-Sadri and Lensmire consid-ered such words and provided an algorithm for generatingall of them using techniques from graph theory. In this pa-per, we exploit their approach in order to count the numberof minimal Sturmian words of order N and we show thatthis number is connected to Euler’s totient function fromnumber theory. We also present some other results thatcome from applying this graph theoretical framework to

52 DM12 Abstracts

subword complexity.

Francine Blanchet-SadriUniversity of North [email protected]

Sean [email protected]

Bob ChenUniversity of California at San [email protected]

CP13

Recurrence in Infinite Partial Words

The recurrence function Rw(n) of an infinite word w wasintroduced by Morse and Hedlund in relation to symbolicdynamics. It is the size of the smallest window such that,wherever its position on w, all length n subwords of wwill appear at least once inside that window. The recur-rence quotient ρ(w) of w, defined as limsup of the quotientof Rw(n) and n, is useful for studying the growth rate ofRw(n). It is known that if w is periodic, then ρ(w) = 1,while if w is not, then ρ(w) ≥ 3. A long standing conjec-ture from Rauzy states that the latter can be improved toρ(w) ≥ (2.5+.5

√5) ∼ 3.618, this bound being true for each

Sturmian word and being reached by the Fibonacci word.In this paper, we study in particular the spectrum of valuestaken by the recurrence quotients of infinite partial words,which are sequences that may have some undefined posi-tions. In this case, we determine exactly the spectrum ofvalues, which turns out to be 1, every real number greaterthan or equal to 2, and∞. More precisely, if an infinite par-tial word w is “ultimately factor periodic”, then ρ(w) = 1,while if w is not, then ρ(w) ≥ 2, and we give constructionsof infinite partial words achieving each value.


Bob ChenUniversity of California at San [email protected]

Sinziana MunteanuPrinceton [email protected]

CP13

Unavoidable Sets

Partial words are sequences over a finite alphabet that maycontain some undefined positions called holes. We firstconsider unavoidable sets of partial words of equal length.We compute the minimum number of holes in sets satisfy-ing some conditions (summed over all partial words in thesets). We also construct sets that achieve this minimum.This is a step towards the difficult problem of fully charac-terizing all unavoidable sets of partial words. We are nextconcerned with the complexity of deciding the avoidabilityof sets of partial words over an arbitrary alphabet. Towardsthis, we investigate the minimum size of unavoidable setsof partial words with a fixed number of holes. Addition-ally, we analyze the complexity of variations on the decisionproblem when placing restrictions on the number of holes

and length of the words.


Laure FlapanYale [email protected]

Steven [email protected]

Elizabeth ReilandHarvey Mudd [email protected]

Stephen WatkinsVanderbilt [email protected]

CP13

Solving Wilf’s Sigma-Tau Problem

We honor Herb Wilf (1931-2012) by solving Exercise 6from his textbook Combinatorial Algorithms with AlbertNijenhuis (1975). The symmetric group Sn is generated byσ = (1 2 · · · n) and τ = (1 2), and Wilf asks if Sn can besequenced ”so that each is obtained from its predecessor”by σ or τ . We settle Wilf’s open problem with a cyclicsequence when n is odd, and a non-cyclic sequence when nis even.

Aaron M. WilliamsUniversity of VictoriaDepartment of Computer [email protected]

CP14

Monotone Path Systems in Regions with Holes

A monotone path system (MPS) is a finite set of pairwisedisjoint paths (polygonal arcs) in the plane such that everyhorizontal line intersects each of the paths in at most onepoint. Consider a simple polygon in the xy-plane whichbounds the polygonal region D. Let T and B be two finite,disjoint, equicardinal sets of points of D. Cameron andSachs gave a polytime algorithm for finding a MPS joiningT with B. We consider polygonal regions with holes.

Kathie CameronWilfred Laurer University, [email protected]

Katie TsujiWilfrid Laurier [email protected]

CP14

Lattice-point Generating Functions for Free Sumsof Convex Sets

The join of two convex sets in Euclidean space is the convexhull of their union. This join is called a free sum under cer-tain conditions on the convex sets. We present a multivari-ate generating function that gives information about thelattice points in the free sum. This work is motivated by

DM12 Abstracts 53

(and recovers) a product formula of Braun for the Ehrhartseries of the free sum of certain lattice polytopes.

Pallavi JayawantBates [email protected]

Matthias BeckSan Francisco State [email protected]

Tyrrell B. McallisterUniversity of [email protected]

CP14

A New Lower Bound Based on Gromovs Methodof Selecting Heavily Covered Points

Classical theorem of Barany states that there exists cd > 0such that for every n-set P of points in Rd in general posi-tion, there exists a point of Rd contained in at least cd

(n

d+1

)

(d + 1)-simplices with vertices at the points of P . Gro-mov improved the known lower bound on cd by topologicalmeans. Using methods from extremal combinatorics, weimprove one of the quantities appearing in Gromov’s ap-proach and thereby provide a new stronger lower bound oncd for arbitrary d.

Lukas MachComputer Science Institute of Charles University (IUUK)Lesser Town Square, no. 25, [email protected]

Daniel KralComputer Science Institute of Charles University (IUUK)[email protected]

Jean-Sebastien SereniCNRS (LIAFA)[email protected]

CP14

On Structures of Geometrically Realizable Trian-gulations on the Mobius Band

Let G be a graph embedded in to a surface F 2. A geometricrealization of G is an embedding of F 2 into a Euclidian 3-space with no self-intersection such that each face of G is aflat polygon. In 1983, Brehm found a Mobius triangulation(i.e., a triangulation on the on the Mobius band) whichhas no geometric realization. In this talk, we characterizegeometrically realizable Mobius triangulations.

Shoichi TsuchiyaTokyo University of [email protected]

CP14

Grid Representation for the Triangulations of theTorus

William Thurston describes all triangulations of the spheresuch that each vertex has degree six or less. In this talk,we present a grid representation for the triangulations ofthe torus with at most two odd vertices where each vertexhas degree five or more. We also discuss the motivation

behind studying this family, and some applications of ourdescription. This is a joint work with Luis Goddyn.

Maryam Verdian RiziSimon Fraser [email protected]

CP15

n-Tournaments That Have n-Integer Signatures

A compact representation for many digraphs is defined ina ”labeled graph” context. Each positive integer of an n-element subset S (called the signature of the digraph) ofZm\{0}, is assigned to a vertex of a digraph with n ver-tices. An arc from a vertex x to another vertex y existsif and only if (y − x) mod m is an element of the signa-ture. A digraph defined by such a signature is called amod difference digraph. In this presentation, we discussthe n-Tournaments Tn having uniquely defined n-Integersignatures.

Vasudeva AcharyaSrinivas Institute of Technology, [email protected]

Suresh M. HegdeNational Institute of Technology [email protected]

CP15

New Characterizations of Proper Interval Bigraphs

The class of proper interval bigraphs was introduced andcharacterized by Sen and Sanyal in terms of monotone con-secutive arrangement of adjacency matrix.In this paperwe define astral triple of edges in a bigraph and charac-terize proper interval bigraph in terms of this and othernotions.Tucker characterized proper circular arc graphs interms of circularly compatible 1’s of adjacency matrices.Wehave shown an inter-relation between monotone consecu-tive arrangement and circularly compatible 1’s.

Ashok K. DasDept. of Pure Mathematics, University of Calcutta. [email protected]

CP15

House of Graphs: a Database of Interesting Graphs

In this talk we will present a new searchable database ofgraphs. Next to complete lists of some graph classes (suchas fullerenes or snarks), also a list of graphs that alreadyturned out to be interesting and relevant in the study ofgraph theoretic problems is offered. We will demonstratehow users can perform queries on this database and howthey can add new interesting graphs to it. House of Graphsis accessible at http://hog.grinvin.org

Jan GoedgebeurGhent [email protected]

Gunnar BrinkmannUniversity of Ghent,[email protected]

54 DM12 Abstracts

Kris CoolsaetGhent [email protected]

Hadrien MelotUniversite de [email protected]

CP15

Immersions of Complete Graphs in Graphs withMinimum Degree n− 1

In 2003, F. Abu-Khzam and M. Langston conjectured thatan n-chromatic graph contains an immersion of a completegraph on n vertices. We will look at an attempt to provethis by looking at a stronger statement involving minimumdegree. We will consider examples that show a graph withminimum degree n − 1 need not contain an immersion ofa complete graph on n vertices for n greater than or equalto 8.

Megan Heenehan, Karen CollinsWesleyan [email protected], [email protected]

CP15

Fat Hoffman Graphs with Smallest Eigenvalue atLeast −1− τIn this talk, we show that all fat Hoffman graphs withsmallest eigenvalue at least −1− τ , where τ is the goldenratio, can be described by a finite set of fat (−1− τ )-irre-ducible Hoffman graphs. In the terminology of Woo andNeumaier, we mean that every fat Hoffman graph withsmallest eigenvalue at least −1−τ is anH-line graph, whereH is the set of isomorphism classes of maximal fat (−1−τ )-irreducible Hoffman graphs.


Akihiro MunemasaTohoku [email protected]

Tetsuji TaniguchiMatsue College of [email protected]

CP15

Finding 1-Factors in Realizations of Degree Se-quences

Kundu proved a necessary and sufficient condition for a de-gree sequence to have a realization containing a k-factor.Brualdi and independently Busch, Ferrara, Hartke, Jacob-son, Kaul and West conjectured that the same conditioncould yield much more: a realization of the sequence con-taining k edge-disjoint 1-factors. In this talk, we describetwo results making partial progress towards this conjecture.

Tyler SeacrestUniversity of Nebraska – Lincolnt [email protected]

CP16

Cayley Factorization and the Area Method

Given a bracket polynomial B, i.e. a polynomial in formaldeterminants. Assume B is homogenous and has integercoefficients. Let e(B,C) be the evaluation of B for a pointconfiguration C. The Cayley factorization problem asks fora synthetic construction for testing e(B,C) = 0. We givean explicit algorithm that generates a bracket monomialM and a construction which tests e(M · B,C) = 0. Thisconstruction is more geometrically motivated and compactthan the one of Sturmfels and Whiteley. M has lower de-gree.

Susanne ApelTechnische Universitaet [email protected]

CP16

Variations on the Erdos Discrepancy Problem

The Erdos Discrepancy Problem asks if there exists a se-quence {ti}i>0 over {−1, 1} and a constant c such that

|∑k

i=1tid| < c for all k, d ∈ N. We examine variations on

this problem where the values of d are restricted to cer-tain subsets of N and certain values of c are fixed. Wecount the number of sequences satisfying these conditions,find the lexicographically least such sequence, and presenta number theory conjecture that arises.

Alexander LeongUniversity of WaterlooSchool of Computer of [email protected]

Jeffrey ShallitUniversity of WaterlooSchool of Computer [email protected]

CP16

1 12-Designs Arising from Group Rings

Let T = (P ,B, I) be a tactical configuration with param-eters (v, b, k, r). For every point x ∈ P and every blockB ∈ B, let φ(x,B) be the number of flags (y, C) ∈ I suchthat y ∈ B \ {x}, x ∈ C and C �= B. A 1 1

2-design with pa-

rameters (v, b, k, r;α, β) is a tactical configuration T suchthat

φ(x,B) =

{α, x /∈ B;β, x ∈ B.

Examples of 1 12-designs include 2-designs, complete bipar-

tite graphs Kn,n, transversal designs, and partial geome-tries. In this talk, we introduce a method which can beviewed as a generalization of difference sets to constructsymmetric 1 1

2-designs. We also give non-existence results

on symmetric 1 12-designs in connection to directed strongly

regular graphs.

Oktay OlmezIowa State [email protected]

CP16

On Iterated Functions, Directed Graphs, and Re-

DM12 Abstracts 55

cursion

In this presentation, we show how enumerations of thebackwards iterates of certain piecewise defined functionslead to general second order linear recurrence relations. Inparticular, for each integer k ≥ 2, we provide a piecewisefunction fk that acts on k equal subdivisions of the unitinterval [0, 1], and the recurrence relation that results fromcounting the solutions of fk(x) = ψ for some ψ in (0, 1).

Christopher Raridan, Elliot KropClayton State [email protected], [email protected]

CP17

Universal H-Colourable Graphs

Rado constructed a (simple) denumerable graph R with thepositive integers as vertex set with the following edges: Forgiven m and n with m < n, m is adjacent to n if n has a 1 inthe m’th position of its binary expansion. It is well knownthat R is a universal graph in the set Ic of all countablegraphs (since every graph in Ic is isomorphic to an inducedsubgraph of R) and that it is a homogeneous graph (sinceevery isomorphism between two finite induced subgraphsof R extends to an automorphism of R). In this talk wediscuss a graph U(H) which is H-universal in →Hc, theinduced-hereditary hom-property of H-colourable graphsconsisting of all (countable) graphs which have a homo-morphism into a given (countable) graph H . If H is the(finite) complete graph Kk, then →Hc is the property ofk-colourable graphs. The universal graph U(H) is char-acterised by showing that it is, up to isomorphism, theunique denumerable, H-universal graph in → Hc whichis H-homogeneous in → Hc. The graphs H for whichU(H) ∼= R are also characterised.

Izak BroereUniversity of [email protected]

Johannes HeidemaUniversity of South [email protected]

CP17

A Generalization of Menger’s Theorem

Consider an acyclic directed network G with sourcesS1, S2, · · · , Sl and distinct sinks R1, R2, · · · , Rl. For i =1, 2, · · · , l, let ci denote the min-cut between Si and Ri.Then, by Menger’s theorem, there exists a group of ciedge-disjoint paths from Si to Ri, which will be referredto as a group of Menger’s paths from Si to Ri in thispaper. Although within the same group they are edge-disjoint, the Menger’s paths from different groups mayhave to merge with each other. In this paper, we provethat by choosing Menger’s paths appropriately, the num-ber of mergings among different groups of Menger’s pathsis always bounded by a constant, which is independent ofthe size and the topology of G. The tightest such con-stant for the all the above-mentioned networks is denotedby M(�∞, �∈, · · · , �∈) when all Si’s are distinct, and byM∗(�∞, �∈, · · · , �∈) when all Si’s are in fact identical. Itturns out that M and M∗ are closely related to the net-work encoding complexity for a variety of networks. Com-putation of these two functions, however, appears to berather difficult; so far there are no explicit formulas forM(�∞, �∈, · · · , �∈) and M∗(�∞, �∈, · · · , �∈) for a generic

parameter c1, c2, · · · , cl. We also derive exact values of andtighter bounds on M and M∗ for some parameters. Seehttp://arxiv.org/abs/0805.4059 for the full manuscript.

Guangyue HanDepartment of MathematicsUniversity of Hong [email protected]

CP17

Another Proof for Lovasz’s Cathedral Theorem

A graph with perfect matchings is satuated if additionof any complement edges creates new perfect match-ings. Cathedral theorem, which is originally investi-gated by Lovasz (1986) and later given another proof bySzigeti (1993), is a constructive charactrization of sat-urated graphs and appears in proving the theorem ontoatally-k-covered graphs. Here, we gave the third prooffor Lovasz’s Cathedral theorem by revealing the orderedstructure of graphs with perfect matchings.

Nanao KitaKeio [email protected]

CP17

Graphs of Small Rank-Width Are Pivot-Minors ofGraphs of Small Tree-Width

We prove that every graph of rank-width k is a pivot-minor of a graph of tree-width at most 2k. We also provethat graphs of rank-width at most 1, equivalently distance-hereditary graphs, are exactly vertex-minors of trees, andgraphs of linear rank-width at most 1 are exactly vertex-minors of paths.

O-Joung KwonDepartment of Mathematical Sciences, [email protected]

Sang-Il [email protected]

CP17

Saturation Numbers for Families of Graph Subdi-visions

For a family F of graphs, a graph G is F-saturated if Gcontains no member of F but the addition of any edge inG results in one. The minimum number of edges in anF-saturated graph of order n is denoted sat(n,F). For agraph H , S(H) denotes the family of graphs resulting fromreplacing the edges of H with internally disjoint paths. Wedetermine sat(n,S(C�)) for small t, and provide a boundwhen t is arbitrarily large. We also examine sat(n,S(K�))in light of a result of Wagner’s on minors.

Craig TennenhouseUniversity of New [email protected]

Michael FerraraUniversity of Colorado [email protected]

Michael Jacobson

56 DM12 Abstracts

University of Colorado, [email protected]

Kevin MilansUniversity of South [email protected]

Paul WengerUniversity of Colorado [email protected]

MS1

Variations on the Majorization Order

The key role played by the majorization (aka dominance)order on integer partitions as a tool in both pure and ap-plied mathematics has been widely recognized and docu-mented. Recently, variants of the classical majorization or-der have appeared in the study of posets defined by certainsymmetric function inequalities. We will describe thesevariants, focusing on differences and similarities betweenthese posets and the classical majorization order.

Curtis GreeneHaverford [email protected]

MS1

Linear Extension Diameter and Reversal Ratio

Define the distance between two linear extensions L1 andL2 of a poset P as the number of incomparable pairs x, ywith x <L1 y and y <L2 x. The linear extension diame-ter of P, led(P) is the maximum distance over all pairs oflinear extensions. We introduce the reversal ratio of P asthe ratio of led(P) to the total number of (unordered) in-comparable pairs of P. We provide a family of posets withreversal ratio tending to 0 with the number of points andinvestigate questions involving bounding the reversal ratiofor posets of fixed width and posets of fixed dimension.

Mitchel T. KellerLondon School of [email protected]

Graham BrightwellDepartment of MathematicsLondon School of [email protected]

MS1

Forbidden Structures for Efficient First-Fit ChainPartitioning

Using natural numbers for identifying chains, First-Fit al-gorithm processes elements of a poset P in some order andfor each element it assigns the smallest natural numbersuch that all elements with the same number form a chain.Let P be a class of posets with bounded-width, closed un-der taking induced subposets. We prove that there is aconstant c such that all posets from P are partitioned byFirst-Fit into at most c chains if and only if there is a width2 poset not in P .

Bartlomiej E. BosekJagellion University, [email protected]

Tomasz Krawczyk, Grzegorz MateckiJagiellonian [email protected], [email protected]

MS1

Forbidden Induced Posets in the Boolean Lattice

The induced Turan number La∗(n,P) is the maximum sizeof a family of elements in the n-dimensional Boolean latticethat does not contain P as an induced subposet. Not muchis known about La∗(n,P) when the Hasse diagram of Pcontains cycles. We present bounds on La∗(n,P) for somesuch posets. This is joint work with Linyuan Lu.

Linyuan Lu, Kevin MilansUniversity of South [email protected], [email protected]

MS1

Title Not Available at Time of Publication

Abstract not available at time of publication

William T. TrotterSchool of MathematicsGeorgia Institute of [email protected]

MS2

Lie Theory for Hyperplane Arrangements

There is a well-known isomorphism of representations be-tween the homology of the partition lattice and the mul-tilinear part of the free Lie algebra. This is a result ofHanlon, Joyal, and Stanley. Rich combinatorics underliesthis fact, as shown by subsequent work of several authors,including Barcelo, Bergeron, Bjorner, Garsia, Reutenauerand Wachs. In this talk we propose an extension of thisand other classical results to real linear hyperplane ar-rangements. A central role is played by the notion of Hopfmonoid. We develop a Lie theory for these objects whichincludes the Cartier-Milnor-Moore and Poincare-Birkhoff-Witt theorems. The classical results are obtained by spe-cializing to the case of braid arrangements. This is jointwork in progress with Swapneel Mahajan.

Marcelo AguiarTexas A&[email protected]

Swapneel MahajanIIT [email protected]

MS2

Tesler Matrices, Parking Functions, and DiagonalHarmonics

We show how the Hilbert series of the space of DiagonalHarmonics can be expressed in terms of combinatorial ob-jects called Tesler matrices. This leads to some open pos-itivity questions involving what we call “colored parkingfunctions”. This is joint work with D. Armstrong, A. Gar-sia, B. Rhoades, and B. Sagan.

Jim HaglundUniversity of PennsylvaniaDepartment of Mathematics

DM12 Abstracts 57

[email protected]

MS2

Combinatorial Ergodicity in Products of Chains

Many cyclic actions τ on a finite set S of combinatorialobjects, along with many natural statistics φ on S, ex-hibit“combinatorial ergodicity’: the average of φ over eachτ -orbit in S is the same as the average of φ over the wholeset S. This phenomenon was first noticed by Panyushev in2007 in the context of antichains in root posets; Armstrong,Stump, and Thomas proved his conjecture in 2011. We de-scribe a theoretical framework for results of this kind, anddiscuss old and new results for products of chains.

Tom Roby

University of Connecticut/[email protected]

James ProppU Mass [email protected]

MS2

Classical and Quasi Symmetric Hall-LittlewoodPolynomials and Transition Matrices

An interesting property of Hall-Littlewood polynomials isthat they interpolate between the Schur and monomialsymmetric functions at t = 0 and 1, respectively. Hiverthas constructed a family of quasisymmetric functions witha parameter t, which interpolate in a similar way betweenthe fundamental and monomial quasisymmetric functions.In this talk we present a combinatorial interpretation forthe expansion of Hall-Littlewood polynomials in terms ofHivert functions, as well as other transition matrices be-tween the latter and other families of quasisymmetric func-tions.

Nicholas LoehrVirginia [email protected]

Luis [email protected]

Gregory WarringtonUniversity of [email protected]

MS2

Demazure Crystals, Kirillov-Reshetikhin Crystals,and the Energy Function

I will discuss a paper with Anne Schilling which surveysand expands upon some relationships between Demazurecrystals of non-exceptional affine Kac-Moody algebras andKirillov-Reshetikhin (KR) crystals. In particular, certainDemazure crystals are isomorphic as classical crystals totensor products of KR crystals, and we show that thisisomorphism intertwines the natural affine grading on theDemazure crystals with a combinatorially defined energyfunction. This leads to a formula for the Demazure char-acter in terms of the energy function, and has applicationsto symmetric function theory since certain specializationsof Macdonald polynomials are equal to specializations of

Demazure characters.

Peter [email protected]

MS3

Conjectures Equivalent to the Borodin-KostochkaConjecture that a Priori Seem Weaker

Borodin and Kostochka conjectured that every graph Gwith maximum degree Δ ≥ 9 satisfies χ ≤ max{ω,Δ− 1}.We carry out an in-depth study of minimum counterexam-ples to the Borodin-Kostochka conjecture. Our main tool isthe classification of graph joins A∗B with |A| ≥ 2, |B| ≥ 2that are f -choosable, where f(v) := d(v)− 1 for each ver-tex v. Since such a join cannot be an induced subgraphof a vertex critical graph with χ = Δ, we have a wealthof structural information about minimum counterexamplesto the Borodin-Kostochka conjecture. Our main result isto prove that certain conjectures that a priori seem weakerthan the Borodin-Kostochka Conjecture are in fact equiva-lent to it. One such equivalent conjecture is the following:Any graph with χ ≥ Δ = 9 contains K3∗K6 as a subgraph.

Daniel CranstonVirginia Commonwealth UniversityDepartment of Mathematics and Applied [email protected]

MS3

List-coloring on Surfaces with Some Small Lists

C. Thomassen’s celebrated 5-list-coloring theorem for pla-nar graphs proves more. Among its consequences are theresults that 1) a planar graph with one vertex with a 1-listand the rest with 5-lists and 2) a planar graph with 3-listson the vertices of one face and 5-lists elsewhere can be list-colored. We study generalizations of these two variationsfor graphs on nonplanar surfaces. We obtain initial resultsfor graphs on the projective plane and torus and begin astudy for surfaces with larger Euler genus.

Alice M. DeanSkidmore CollegeMathematics and Computer Science [email protected]

Joan P. HutchinsonMacalester CollegeDepartment of [email protected]

MS3

Fractional Colorings of Cubic Graphs

Fractional coloring is a relaxation of classical graph col-oring. Each independent set is assigned weight and eachvertex is required to be in independent sets of weight atleast one. The goal is to minimuze the total weight of in-dependent sets. In the talk, we present two recent results:The fractional chromatic number of any triangle-free cu-bic graph is at most 32/11 and the fractional chromaticnumber of any cubic graph with sufficiently large girth issmaller than 23/10.

David FergusonLondon School of Economics

58 DM12 Abstracts

[email protected]

Tomas KaiserUniversity of West [email protected]

Frantisek KardosINRIA [email protected]

Daniel KralCharles [email protected]

Jan VolecIUUK Charles [email protected]

MS3

4-critical Graphs on Surfaces without ContractibleCycles of Length at Most 4

We show that every 4-critical graph G embedded in a fixedsurface Σ so that every contractible cycle has length atleast 5 can be expressed as G = G′ ∪ G1 ∪ G2 ∪ . . . ∪ Gk,where |V (G′)| and k are bounded by a constant (dependinglinearly on the genus of Σ) and G1, . . . , Gk are graphs (ofunbounded size) whose structure we describe exactly. Theproof is computer-assisted— we use computer to enumerateall plane 4-critical graphs of girth 5 with a precolored cycleof length at most 16, that are used in the basic case of theinductive proof of the statement.

Zdenek DvorakDepartment of Applied MathematicsCharles University, [email protected]

Bernard LidickyUniversity of Illinois, Urbana, [email protected]

MS3

Extending Graph Choosability Results toPaintability

Introduced independently by Schauz and by Zhu, theMarker/Remover game is an on-line version of list col-oring. The resulting graph parameter, paintability, is atleast the list chromatic number (the choosability). Westrengthen several choosability results to paintability. Westudy paintability of joins with complete graphs and de-termine bounds on the paintability of complete bipartitegraphs. We characterize 3-paint-critical graphs and showthat claw-free perfect graphs with ω(G) ≤ 3 have paintabil-ity equal to chromatic number. Finally, we introduce andstudy sum-paintability, the analogue of sum-choosability.

James CarraherUniversity of [email protected]

Sarah Loeb, Thomas Mahoney, Gregory Puleo, Mu-TsunTsaiUniversity of [email protected], [email protected],[email protected], [email protected]

Douglas B. WestUniversity of Illinois, UrbanaDepartment of [email protected]

MS4



Hamed HatamiDepartment of MathematicsPrinceton [email protected]

MS4

On a Problem of Erdos and Rothschild on Edges inTriangles

Erdos and Rothschild asked to estimate the maximumnumber h(n, c) such that every n-vertex graph with at leastcn2 edges, each of which is contained in at least one trian-gle, must contain an edge that is in at least h(n, c) triangles.In 1987, Erdos asked whether for every c > 0, there is ε > 0such that h(n, c) > nε. We prove h(n, c) = nO(1/ log log n)

for every fixed c < 1/4, the first improvement over an oldconstruction of Alon and Trotter.

Jacob [email protected]

Po-Shen LohDepartment of Mathematical SciencesCarnegie Mellon [email protected]

MS4

Turan’s Brickyard Problem and Flag Algebras

Recently, Razborov developed a ”flag algebra” calculuswhich captures in pure form many of the techniques usedin extremal combinatorics. This calculus has been success-fully used in the last few years to improve the best knownbounds on many of the long-standing open problems inextremal graph theory. In this talk we describe an appli-cation to the Turan’s brickyard problem: the problem ofdetermining the crossing number of the complete bipartitegraph Km,n. Beas on joint work with Yori Zwols.

Sergey NorinMcGill [email protected]

MS4

Turan Densities of Hypergraphs Related to Kkk+1

Let B(k)i be the k-uniform hypergraph on the vertex set

S ∪ T with |S| = i and |T | = k − 1 with the edges consist-ing of all k-sets containing S or T . We derive upper and

lower bounds for the Turan density of B(k)i that are close to

each other as k→∞. We also obtain asymptotically tightbounds for the Turan density of several other infinity fam-ilies of hypergraphs. The construction that supports thelower bounds is derived from elementary number theoryby probabilistic arguments. The upper bounds are derived

DM12 Abstracts 59

from the results of de Caen, Sidorenko, and Keevash.

Yi ZhaoGeorgia State [email protected]

Jozsef BaloghUniversity of [email protected]

Tom BohmanCarnegie [email protected]

Bela BollobasTrinity College, CambridgeUniversity of [email protected]

MS4

Extremal results in sparse pseudorandom graphs

Szemeredi’s regularity lemma is a fundamental tool in ex-tremal combinatorics. However, the original version isonly helpful in studying dense graphs. In the 1990s, Ko-hayakawa and Rodl proved an analogue of Szemeredi’s reg-ularity lemma for sparse graphs as part of a general pro-gram toward extending extremal results to sparse graphs.Many of the key applications of Szemeredi’s regularitylemma use an associated counting lemma. In order toprove analogues of these results for sparse graphs, it re-mained a well-known open problem to prove a countinglemma in sparse graphs. The main advance of this paperlies in a new counting lemma, proved following the func-tional approach of Gowers, which complements the sparseregularity lemma of Kohayakawa and Rodl, allowing us tocount small graphs in regular subgraphs of a sufficientlypseudorandom graph. We use this to prove analogues ofseveral well-known combinatorial theorems, including thegraph removal lemma, the Erdos-Stone-Simonovits theo-rem, and Ramsey’s theorem, for subgraphs of sparse pseu-dorandom graphs. These results extend and improve upona substantial body of previous work.

David ConlonUniversity of [email protected]

Jacob [email protected]

Yufei ZhaoMassachusetts Institute of [email protected]

MS5

On the Directed Oberwolfach Problem

The Oberwolfach problem asks whether the Kn (or Kn− Iif n is even) admits a 2-factorization in which each factoris isomorphic to a specified 2-factor F . We consider a di-rected variant of this problem, in which we are now requiredto factorize the complete symmetric digraph K∗

n. In par-ticular, we consider the question of determining necessaryand sufficient conditions for the existence of a resolvabledecomposition of K∗

n into directed m-cycles, and present

some results towards its solution.

Andrea BurgessRyerson [email protected]

Patrick [email protected]

Mateja SajnaUniversity of [email protected]

MS5

Defining Sets in Combinatorial Arrays

A defining set in a combinatorial array is a partially filled-in array with a unique completion. We compare what isknown about the defining sets of (0, 1)-matrices and Latinsquares. While these arrays on the surface seem quite dif-ferent, for certain cases the minimum possible size for adefining set appears to be the same.

Nicholas CavenaghUniversity of [email protected]

MS5

Graph Decompositions and Convexity

One of the most interesting conjectures in the area of graphdecompositions is due to Nash-Williams. It states thatthe edge set of any sufficiently large graph with all evendegrees, 3m edges, v vertices and minimum degree of atleast 3/4v can be partitioned into triangles. We discuss anew approach for exploring this problem using the inclusionmatrix of the Johnson scheme and the facets of the convexcone it generates.

Kseniya GaraschukUniversity of [email protected]

Peter DukesAssistant [email protected]

MS5

Extending the Bruck-Ryser-Chowla Theorem toCoverings

The celebrated Bruck-Ryser-Chowla theorem rules out theexistence of various balanced incomplete block designs, in-cluding projective planes of infinitely many orders. In thistalk I will discuss how (parts of) the Bruck-Ryser-Chowlatheorem can be extended to establish the nonexistence ofvarious covering designs and, consequently, the nonexis-tence of certain decompositions of a complete graph intocopies of a smaller complete graph minus an edge.

Daniel HorsleyMemorial University, Newfoundland, [email protected]

Darryn Bryant, Melinda Buchanan, Barbara Maenhaut,Victor ScharaschkinUniversity of Queensland

60 DM12 Abstracts

[email protected], [email protected],[email protected], [email protected]

MS5

Trinal Decompositions of Steiner Triple Systems

Let STS(n) be a Steiner triple system of order n. A tri-angle T is a set of three pairwise intersecting triples of anSTS(n) whose intersection is empty. Z. Furedi posed aquestion whether the set of triples of any STS(n) can bedecomposed into triangles. This question remains largelyunanswered, although examples of such decomposition areknown for every admissible order n ≡ 1 or 9 (mod 18). Atriangle T = {{a, b, c}, {c, d, e}, {e, f, a}} in an STS(n) issometimes called a hexagon triple because the outer edgesab, bc, cd, de, ef , fa form a hexagon. Depending on agraph-theoretic or geometric representation, respectively,that triangle determines naturally two more triples: theinner triple {a, c, e} and the midpoint triple {b, d, f}. Ineither case, the number of inner triples (called type 1) orthat of midpoint triples (type 2) equals one third of the to-tal number of triples in an STS(n). A problem which willbe discussed concerns the existence of three distinct decom-positions of an STS(n) into triangles such that the unionof three collections of type 1 triples (type 2, respectively)from these three decompositions form a set of triples of aSteiner triple system of the same order n. Such decomposi-tions are called trinal decompositions of type 1 and type 2,respectively. Solutions to the existence question for trinaldecompositions of type 1 and type 2 will be presented.

Mariusz MeszkaAGH University of Science and [email protected]

Charles C. LindnerAuburn [email protected]

Alexander RosaDepartment of MathematicsMcMaster [email protected]

MS6

Structural Requirements for RNA Elements

RNA plays a major role in many regulatory circuits of cells,and new and often suprising function are discovered con-tinously, The structure of RNA elements determines theassociated function for many ncRNAs. Clustering by sim-ilarity on the level of sequence and structure is now anaccepted approach for annotating ncRNAs and for deter-mining new ncRNA classes. The problem, however, is thelarge number of RNAs to be clustered (e.g., up to 450.000predicted ncRNAs in human), and the complexity of thesimilarity test. We have introduced a new alignment-freemethod for clustering that avoids the complex all-againts-all pairwise comparisons, and is still able to cluster accord-ing to sequential and structural properties. Furthermore,we will discuss application problems like the bacterial adap-tive immunesystem called CRISPR that is based on RNA.

Rolf BackofenUniversity of [email protected]

MS6

Understanding SHAPE-directed RNA SecondaryStructure Prediction

We investigate the interplay between experimental dataand thermodynamic model via stochastic simulations.RNA prediction under the NNTM is a discrete optimiza-tion problem whose accuracy can be improved by the in-corporation of auxiliary information. However, our resultsdemonstrate that improvements in SHAPE-directed pre-dictions are non-uniform and correlated with original MFEaccuracies. This analysis suggests that further advancesin computational molecular biology are needed to reliablypredict RNA secondary structures.

Zsuzsanna SukosdNanoscience CenterAarhus [email protected]

Christine E. HeitschSchool of MathematicsGeorgia [email protected]

MS6

Detecting SNP-Induced Structural Changes inRNA: Application to Disease Studies

Single Nucleotide Polymorphisms (SNPs) are often linkedto critical phenotypes such as diseases. However, the spe-cific molecular mechanisms by which a causal SNP acts isusually not obvious. Changes in RNA secondary structureemerge as a possible explanation. We introduce remuRNA,an efficient method to compute the distance between thenative and mutant RNA structure Boltzmann ensembles.We applied remuRNA to determine which of the disease-associated non-coding SNPs are potentially related to RNAstructural changes.

Raheleh SalariStanford [email protected]

Teresa [email protected]

MS6

Efficient Algorithms to Explore the RNA Muta-tional Landscape

Understanding the relationship between RNA sequencesand structures is essential to decipher evolutionary pro-cesses, predict deleterious mutations and design syntheticmolecules. We introduce RNAmutants, the first algorithmfor exploring complete RNA sequence-structures maps inpolynomial time and space. Using statistical mechanicsand importance sampling techniques, we estimate the ther-modynamical pressure applied on sequences and explore re-gions of the mutational landscape preserving the nucleotidecomposition. We show that C+G-contents influence theevolutionary accessible structural ensembles.

Jerome WaldispuhlMcGill [email protected]

Bonnie Berger

DM12 Abstracts 61

Applied Mathematics, [email protected]

Srinivas DevadasCSAIL, [email protected]

Peter CloteBiology, Boston [email protected]

Yann PontyLIX, Ecole [email protected]

MS7

Path Graphs, PR-trees, and Split Decomposition

In this work we preset new characterizations of (directed)path graphs*. First, we describe a new data structure (PR-trees) which captures the set of path-tree models of a pathgraph (generalizing PQ-trees). This allows us to character-ize and recognize (directed) path graphs via split decompo-sition. The recognition algorithm runs in O(|V| ∗ |E|) forboth directed path graphs and path graphs. * The inter-section graphs of (directed) paths in (directed) trees.

Steven ChaplickDepartment of Computer Science,University of [email protected]

MS7

Algorithms for Unipolar and Generalized SplitGraphs

G=(V1 ∪ V2, E) is unipolar when V1 is a clique and V2

induces the disjoint union of cliques. A generalized splitgraph is either unipolar or its complement is unipolar. Wepresent an O(nm′)-time recognition algorithm for unipo-lar graphs, where m′ is the number of edges in a minimaltriangulation of the given graph. We efficiently solve fourclassic optimization problems on unipolar and generalizedsplit graphs and prove that perfect code is NP-completefor unipolar graphs.

Elaine M. EschenLane Department of CSEEWest Virginia [email protected]

Xiaoqiang WangWest Virginia [email protected]

MS7

Recognizing Even-Cycle Matroids

A matroid M is an even-cycle matroid if the cycles of Mcorrespond to the even cycles of a signed graph. The prob-lem of recognizing whether a binary matroid (representedby a 0,1 matrix) is an even-cycle matroid is open. Progresson this problem has been hampered by the fact that evencycle matroids can have an arbitrary number of pairwiseinequivalent representations (two signed graph are equiva-lent if they are related by a sequence of Whitney-flips andsignature exchanges). We discuss how to proceed for therecognition problem for the case where we are given a fixed

size minor that is not a projection of a graphic matroid.

Bertrand GueninUniversity of [email protected]

Irene PivottoSimon Fraser University, [email protected]

Paul WollanDepartment of MathematicsUniversity of [email protected]

MS7

Fixed Weight De Bruijn Graphs

Let S be the set of all binary strings of length n with weight(number of 1s) w and w-1. The de Bruijn graph for S isa directed graph G(S) whose nodes are the length n−1prefixes and suffixes of the strings in S. There is an arclabeled x ∈ {0, 1} from α = a1 · · · an−1 to β = a2 · · · an−1xif αx ∈ S. In addition to considering constructions of fixedweight de Bruijn cycles, we consider the problem of deter-mining the diameter of the de Bruijn graph G(S).

Joe SawadaUniversity of GuelphDept. of Computer [email protected]

MS7

Induced and Distance-k Matchings and Some Re-lated Min-Max Relations

A distance-k matching in a graph is a matching such thatthe distance between any two edges of the matching is atleast k. Computing the size of a largest induced matching(distance-2 matching) in a graph is NP-hard. We considerfinding a largest distance-k matching in certain classes ofgraphs. In some cases, along comes a relation betweenthe size of a largest distance-k matching and the size of asmallest appropriate cover.

R Sritharan, Arthur BuschThe University of [email protected], [email protected]

Feodor F. DraganKent State UniversityDept. of Computer [email protected]

Chandra KrishnamurthyThe University of [email protected]

MS8

Q2-Free Families in the Boolean Lattice

For a family of subsets of {1, . . . , n}, ordered by inclusionand a partially-ordered set P , we say that the family is P -free if it does not contain a subposet isomorphic to P . Weare interested in finding ex(n, P ), the largest size of a P -free family of subsets of [n]. It is conjectured that, for any

fixed P , this quantity is (k+o(1))(n2

)for some fixed integer

62 DM12 Abstracts

k, depending only on P . Recently, Boris Bukh has verifiedthe conjecture for P which are in a “tree shape’. Thereare some other small posets P for which the conjecturehas been verified. The smallest for which it is unknown isQ2, the Boolean lattice on two elements. We will discussimprovements of the bounds of Griggs, Li, and Lu. Thisis joint work with Ryan Martin, Iowa State University andMichael Young, Iowa State University.

Lucas J. Kramer, Ryan R. Martin, Michael YoungIowa State [email protected], [email protected],[email protected]

MS8

First-Fit Coloring of Ladder-Free Posets

Bosek and Krawczyk provided a subexponential bound forthe on-line chain partitioning of posets of width w andobserved the problem could be reduced to First-Fit chainpartitioning of 2w2-ladder-free posets of width w. We pro-vide a subexponential upper bound (in terms of w with mfixed) for the performance of First-Fit chain partitioningon m-ladder-free posets. With the Bosek-Krawczyk obser-vation, this yields an slightly improved upper bound forthe general problem.

Matt E. SmithArizona State [email protected]

H. A. KiersteadMathematics DepartmentArizona State [email protected]

MS8

Dimension and Height for Posets with PlanarCover Graphs

Planar posets can have arbitrarily large dimension. How-ever, we show that the dimension of a planar poset isbounded as a function of its height. More precisely, weshow that for each integer h ≥ 2, there exists a least pos-itive integer ch so that if P is a poset with a planar covergraph and height h, then the dimension of P is at mostch. Trivially, c1 = 2. Felsner, Li and Trotter showed thatc2 = 4, but their proof techniques do not seem to applywhen h ≥ 3. In this paper, we establish the existence ofch, although we suspect that the upper bound provided byour proof is far from best possible. From below, a con-struction of Kelly is modified to show that ch is at leasth+ 2.

Noah StreibGeorgia Institute of [email protected]


MS8

The Width of the Family of Maximum Antichains

Felsner, Krawczyk and Micek conjectured in 2010 that themaximum possible width of the family of maximum an-

tichains in a (k + k)-free poset of width w is (k − 1)w−1.This is easily shown to be true for w ≤ 2. We give theproof for w = 3 and present several different constructionsattaining the conjectured extremal value for w ≥ 3. De-spite this support we still feel to be far away from the prooffor all w.

Micha�l Lason, Piotr MicekJagiellonian [email protected], [email protected]



Bartosz WalczakJagiellonian [email protected]

MS8

The Dimension of Posets with Planar CoverGraphs

We study the conditions that bound the dimension ofposets with planar cover graphs. We show that if P isa poset with a planar comparability graph, then the di-mension of P is at most four. We also show that if P hasan outerplanar cover graph, then the dimension of P is atmost four. Finally, if P has an outerplanar cover graphand the height of P is two, then the dimension of P is atmost three. These three inequalities are all best possible.

Stefan FelsnerTechnical University of BerlinInstitute for [email protected]


Veit WiechertTU - [email protected]

MS9

Tropisms, Surfaces and the Puiseux Series

We present a polyhedral method to develop Puiseux seriesexpansions for surfaces, defined by a system of polynomi-als. Our starting point is the construction of cones of nor-mal vectors to the Newton polytopes, associated with thesystem of polynomials. Generating vectors of the normalcone, which lead to an initial form system with regular,isolated solutions are considered pretropisms. When thegenerating vectors of the normal cone lead to an exact, fi-nite representation of the surface or to its Puiseux seriesexpansion, then they are referred to as tropisms.

Danko Adrovic, Jan VerscheldeDepartment of Mathematics, Statistics and ComputerScience

DM12 Abstracts 63

University of Illinois at [email protected], [email protected]

MS9

Divisors on Tropical Varieties

I will present notions of divisors, linear equivalence, andrank on tropical models. A tropical model is a simplicialcomplex together with some numerical data telling how theaffine linear structures relate. Although the motivationscome from algebraic varieties, I will discuss the combina-torial aspects.

Dustin CartwightDepartment of MathematicsYale [email protected]

MS9

Orbits of Projective Point Configurations

Given an r×n matrix, thought of as representing n pointsin projective (r−1)-space, we consider the closure of the or-bit of all projectively equivalent matrices. I will discuss theequations cutting out this variety, its finely graded Hilbertseries, and their relation to the matroid of the point con-figuration.

Alex FinkNorth Carolina State [email protected]

Andrew BergetUniversity of California, [email protected]

MS9

Log-concavity of Characteristic Polynomials andTropical Intersection Theory

In a recent joint work with June Huh, we proved the logconcavity of the characteristic polynomial of a realizablematroid by relating its coefficients to intersection numberson an algebraic variety and applying an algebraic geomet-ric inequality. In this talk, we outline that proof whichinvolves algebraic geometric positivity.

Eric KatzUniversity of Texas [email protected]

MS9

Tropical Torelli Space and Tropical Period Map-ping

I will discuss the Teichmuller type approach to the mod-uli of tropical curves. We also define the tropical periodmapping and investigate the injectivity of this map on theTorelli space (i.e. the generalized Torelli problem). Wewill describe the image of the period map (i.e. the Schot-tky locus). To better understand the Torelli problem andthe Schottky locus, we consider the (marked) moduli of“metric” regular matroids. Various compactifications ofthe moduli of tropical curves will be obtained from thispoint of view.

Farbod ShokriehGeorgia Institute of Technology

[email protected]

MS10

Globally Fair Stable Matchings

Stable marriage instances generally have many stablematchings. In spite of this, Gale and Shapley’s famousalgorithm for computing stable matchings can only out-put two kinds: the man-optimal/woman-pessimal or thewoman-optimal/man-pessimal stable matchings. This hasmotivated the study of fair stable matchings. This talk isabout stable matchings whose fairness is derived from thefact that they are ”good” representatives of the distributivelattice of stable matchings of the instance.

Christine T. ChengUniversity of [email protected]

MS10

Stable Matching as a Heuristic for Cost-BasedMatching Problems

Minimum-cost assignment problems arise in many situa-tions in practice. The optimal solution of such problemsusually requires network flow techniques, which can becomputationally taxing for very large problem instances.Here, we discuss potential heuristics for minimum-cost as-signment problems based on stable matchings, which cangenerally be computed much more efficiently. A numberof approaches will be discussed, along with the results ofempirical testing to assess their viability in practice.

Brian C. Dean, John DabneyClemson [email protected], [email protected]

MS10

Some Open Problems in Matchings with Prefer-ences

Matching problems involving preferences have been stud-ied for many years, and have a range of real-world appli-cations. The stable marriage problem is perhaps the mostfamous such problem, but many variants of it have beenstudied, involving extensions and variations of the basicmodel. Despite the extensive literature in this area, manychallenging (and fascinating) open problems remain. I willsurvey some of these open problems that have connectionswith discrete mathematics, algorithms and complexity.

David F. ManloveUniversity of [email protected]

MS10

A Unified Approach to Equivalence Results in Ob-ject Allocation

We consider probabilistic mechanisms that allocate indi-visible objects to agents by hierarchical exchange using thetop-trading cycles algorithm. The main result is a generaltechnique for proving that seemingly different probabilisticmechanisms are in fact equivalent. This approach simpli-fies and unifies several equivalence results in the literature.The same technique is used to generalize these results tomechanisms in which the priority structure for each ob-ject is given by a tree (instead of a linear ordering of the

64 DM12 Abstracts

agents).

Jay SethuramanColumbia [email protected]

Thiam [email protected]

MS11

Homeomorphically Irreducible Spanning Trees

Let G be a graph. A spanning tree of G is called a homeo-morphically irreducible spanning tree (HIST) if it does notcontain vertices of degree 2. In 1979, Albertson, Berman,Hutchinson, and Thomassen asked the following two ques-tions:

1. Does every triangulation of a surface contain a HISTexcept the triangle?

2. Does every graph with every edge on two trianglescontain a HIST?

We have confirmed both questions positively. The outlinesof the proofs will be given in this talk.

Guantao ChenDepartment of Mathematics and StatisticsGeorgia State [email protected]

Songling ShanGeorgia State [email protected]

MS11

Decompositions of (Hyper)Graphs into Cliques orBicliques

This talk will be a brief survey of recent results regard-ing edge-decompositions of (hyper)graphs into cliques orbicliques. These include various extensions of De Bruijn-Erdos Theorem and Graham-Pollak Theorem. Despite re-cent progress, many problems remain open.

Sebastian CioabaUniversity of Delaware at [email protected]

MS11

Multipartite Version of the Alon-Yuster Theorem

In this talk, we prove the asymptotic multipartite versionof the Alon-Yuster theorem. That is, if k ≥ 3 is an integer,H is a k-colorable graph and γ > 0 is fixed, then for suf-ficiently large n and for every balanced k-partite graph Gon kn vertices with each of its corresponding

(k2

)bipartite

subgraphs having minimum degree at least k−1kn+ ηn, the

graph G has a subgraph consisting of �n/|V (H)|� vertex-disjoint copies of H .

Ryan R. MartinIowa State [email protected]

Jozef SkokanLondon School of [email protected]

MS11

An Erdos-Stone Theorem for Hypergraphs

This talk is about complete k-partite subgraphs in hyper-graphs with sufficiently many k-cliques. These results gen-eralize the Erdos-Stone theorem for 2-graphs.

Vladimir NikiforovDepartment of Mathematical Sciences, University ofMemphisMemphis, TN 38152-3240 , United [email protected]

MS11

On Independent Sets in Steiner Systems

For a hypergraph H , let α(H) denote the largest set ofvertices containing no edge of H . In this talk we give near-optimal bounds on the size of a largest independent set inSteiner systems on n points. We conjecture that the mini-mum value of α(H) over all Steiner systems H on n verticesis asymptotic to

√3n log n as n →∞. The methods are a

combination of spectral techniques and probabilistic com-binatorics, which are useful for various other problems onindependent sets in hypergraphs.

Jacques VerstraeteUniversity of California-San [email protected]

MS12

Graph Homomorphisms: Mixing and Homotopies

Given a k-colouring of a graph, consider the process ofrecolouring a single vertex to obtain a new k-colouring.Provided k is sufficiently large, one can generate all k-colourings of the graph. We study circular colourings giv-ing bounds on how large k/q must be to ensure all circularcolourings can be generated by recolouing. We connect ourwork to discrete homotopies of graph homomorphisms andpre-colourings of graphs.

Richard BrewsterDepartment of Mathematics and StatisticsThompson Rivers [email protected]

Jon NoelMcGill [email protected]

MS12

Defective Colorings and Colorings that AvoidLarge Monochromatic Components

We consider two types of colorings. Those that avoidmonochromatic graphs with large degree and those thatavoid monochromatic graphs with large components. Wediscuss relations between these two parameters and makeseveral remarks on issues of computational complexity.

John GimbelUniversity of Alaska [email protected]

MS12

Kempe-equivalence Classes for 3-edge-colored Cu-

DM12 Abstracts 65

bic Graphs

An edge-Kempe change switches the colors in a maximaltwo-colored chain in a proper edge coloring of a graph. Twoproper edge colorings of a graph are called Kempe equiva-lent if you can get from one to the other by a sequence ofedge-Kempe changes. This work examines Kempe equiv-alence in 3-edge colorable cubic graphs. Not all properedge colorings of a 3-edge colorable cubic graph are Kempeequivalent. We give results about the number of non-equivalent proper edge colorings there can be for certainclasses of cubic graphs.

Ruth HaasSmith [email protected]

Sarah-Marie BelcastroSmith CollegeHampshire College Summer Studies in [email protected]

MS12

Distinguishing Edge Colourings of Graphs

An edge colouring of a graph G is distinguishing providedthat the identity is the only automorphism of G that pre-serves the edge colours. The minimum number of coloursrequired to produce such a colouring is the edge distin-guishing chromatic number of G. In this talk I will describeresults that we have obtained concerning the edge distin-guishing chromatic number and the more general edge dis-tinguishing number of a graph.

Karen SeyffarthMathematics and StatisticsUniversity of [email protected]

Richard BrewsterDepartment of Mathematics and StatisticsThompson Rivers [email protected]

Stacey LamontDepartment of Mathematics and StatisticsUniversity of [email protected]

MS12

Obstructions to Homomorphisms Involving theGraft Extension

Let H1 be a digraph that has an X-enumeration, say{h1, h2, . . . , hn}. Let H2 be a digraph such that H2-colouring is polynomial. Form a new digraph H by deletingthe vertex hn from H1 and replacing it by the digraph H2:every vertex hi ∈ V (H1) that is adjacent to (from) hn isnow adjacent to (from) every vertex in H2. The digraphH is denoted by graft(H1,H2). Given a digraph H , a dual(or obstruction) of H is a digraph F such that G �→ H ifand only if F → G. In this talk we aim to identify theobstructions of H in the case where H = graft(H1,H2).

Jacobus SwartsMathematicsVancouver Island [email protected]

Gary MacgillivrayMathematics and StatisticsUniversity of [email protected]

MS13

TBD on Genome Assembly Algorithms


Max AlekseyevUniversity of South [email protected]

MS13

De Bruijn Graph Based Genome Assembly for Sin-gle Cells

Characterization of environmental bacteria, the vast ma-jority of which elude cultivation, is necessary for many ap-plications including discovery of rare novel species. Theability to generate high-quality draft genome assembliesthat support annotation of the majority of genes will driveadvances in characterizing such uncultured organisms. Re-cent advances in DNA amplification technology have en-abled whole genome sequencing directly from individualcells without requiring growth in culture. These meth-ods amplify femtograms of DNA extracted from a singlebacterial cell into micrograms of DNA needed for currentsequencing platforms. Based on these advances, we devel-oped a specialized software tool for assembling sequencingreads from single cells and applied it to assembly of twoknown genomes, E. coli and S. aureus, and an unknownmarine genome, SAR324 Deltaproteobacterium. Thesedraft de novo single cell assemblies, with no efforts to closegaps and resolve repeats, identify more than 90% of genes.DNA amplification using Multiple Displacement Amplifi-cation (MDA), however, creates serious coverage bias andchimeric fragments. Our algorithm successfully deals withthese errors. Our assembler is based on the celebrated deBruijn graph in which nodes are (k−1)-mers and edges arek-mers. We present the general de Bruijn graph sequenceassembly algorithm as well as our modifications to it.

Hamidreza ChitsazWayne State [email protected]

MS13

Quantifying Uniformity of Mapped Reads

We describe a tool for quantifying the uniformity ofmapped reads in high-throughput sequencing experiments.Our statistic directly measures the uniformity of both readposition and fragment length, and we explain how to com-pute a p-value that can be used to quantify biases arisingfrom experimental protocols and mapping procedures. Ourmethod is useful for comparing different protocols in exper-iments such as RNA-Seq.

Valerie Hower, Richard Starfield, Adam RobertsUniversity of California, [email protected], [email protected],[email protected]

Lior PachterUniversity of California, BerkeleyDept. of Mathematics

66 DM12 Abstracts

[email protected]

MS13

Combinatorial Designs for Sequencing Pooled Sam-ples

Rare variants account for missing heritability that cannotbe explained by common variants. DNA Pooling is a costeffective method to sequence a large number of samples andto discover rare variants. In this talk, I will present two newcombinatorial designs for overlapped pooling strategies andshow their effectiveness in discovering rare variants and inidentifying variant carriers.

Wenhui Wang, Xiaolin Yin, Matthew Hayes, Yoon SooPyon, Jing LiCase Western Reserve [email protected], [email protected], [email protected],[email protected], [email protected]

MS13

Spaced Seeds and their Application in Next Gen-eration Sequencing

We will briefly overview the spaced seeds technique forimproving the speed and sensitivity of homology search.Then, a variation of spaced seeds is introduced to providefull sensitivity reads mapping for next generation sequenc-ing analysis.

Bin MaUniversity of [email protected]

MS14

The Pebbling Threshold of Graph Sequences

A pebbling threshold for a family of graphs is a functiong(n) such that any pebbling configuration of size muchlarger than g(n) is almost surely solvable and of size muchless than g(n) is almost surely unsolvable. We will dis-cuss pebbling threshold for an arbitrary graph family andpresent the results for particular families of graphs and dis-cuss open problems. We will also introduce a probabilisticversion of Graham’s pebbling conjecture for the threshold.

Airat BekmetjevMathematics DepartmentHope [email protected]

MS14

Complexity of Diameter Two Graph Pebbling

We present tight bounds on the number of vertices with twoand three pebbles that can exist in an unsolvable configu-ration on a diameter two graph in terms of the size of thegraph. We use the construction from this result to provethat determining reachability of a vertex is NP-complete,even in graphs of diameter two.

Charles Cusack, Timothy Lewis, Daniel SimpsonDepartment of MathematicsHope [email protected], [email protected],[email protected]

Samuel Taggart

Department of MathematicsOberlin [email protected]

MS14

Graph Pebbling: Past, Present, Future

In this talk we aim to give a gentle introduction to the sub-ject of graph pebbling, including its history from combina-torial number theory, its growth as a purely graph theoretictopic, and its development into an area of combinatorialoptimization. We shall describe some of its variations aswell, such as optimal pebbling, cover pebbling, fractionalpebbling, pebbling thresholds, and rubbling, along with itsconnections to computer science. Lastly, we will presenta few of the major techniques in the field and offer sometantalizing open problems for further research.

Glenn HurlbertArizona State [email protected]

MS14

Graph Rubbling

Graph rubbling is a version of graph pebbling where anadditional move is allowed. The new move adds one peb-ble at vertex u in exchange for removing one pebble eachfrom two vertices that are neighbors of u. Rubbling num-bers are defined analogously to pebbling numbers. Thepurpose of this talk is to give a summary of what is knownabout rubbling and optimal rubbling and to present a setof unanswered questions.

Nandor SiebenDepartment of MathematicsNorthern Arizona [email protected]

MS14

Pebbling Graphs of Diameter Three and Four

Given a configuration of pebbles on the vertices of a con-nected graph G, a pebbling move is defined as the removalof two pebbles from some vertex and the placement of oneof these on an adjacent vertex. The pebbling number of agraph G is the smallest integer k such that for each vertexv and each configuration of k pebbles on G there is a se-quence of pebbling moves that places at least one pebbleon v. We improve on a bound of Bukh by showing thatthe pebbling number of a graph of diameter three on n ver-tices is at most �3n/2+2�, and this bound is best possible.Further, we will describe an asymptotic bound for graphsof diameter four and an asymptotic bound for pebblinggraphs of arbitrary diameter obtained via discharging.

Carl YergerDepartment of MathematicsDavidson [email protected]


Luke PostleGeorgia [email protected]

DM12 Abstracts 67

MS15

Phase Transition in Random Graph Processesthrough the Lens of PDE and Singularity Analy-sis

Erdos and Renyi show that in the standard random graphon n vertices, a phase transition takes place when the num-ber of edges reaches n/2 and a giant component emerges.Since this seminal work, various random graph processeshave been studied. In this talk we discuss new approachesto study the size and structure of components near the crit-ical point of random graph processes: key techniques are aquasi-linear PDE and its singularity analysis.

Mihyun KangTechnische Universitat GrazInstitut fur Optimierung und Diskrete [email protected]

MS15

Self-similarity of Graphs

An old problem of Jacobson and Schonheim asks for themaximum s such that every m-edge graph contains a pairof edge-disjoint isomorphic subgraphs with s edges. Weprove that every m-edge graph contains a pair of edge-disjoint isomorphic subgraphs with at least c(m logm)2/3

edges. We also construct graphs that show this estimateis correct up to a constant factor. Our results improvebounds of Erdos, Pach, and Pyber from 1987.

Choongbum LeeUniversity of California, Los [email protected]

Po-Shen LohDepartment of Mathematical SciencesCarnegie Mellon [email protected]

Benny SudakovDepartment of [email protected]

MS15

Hunting the k-SAT Threshold

We prove that the threshold for the existence of solutionsin random k-NAESAT is 2k−1 ln 2− ( ln 2

2+ 1

4) + εk, where

|εk| ≤ 2−(1−ok(1))k. According to deep but non-rigorousarguments from statistical mechanics, the insufficiency ofcurrent methods to prove such results is due to a changein the geometry of the set of solutions called condensationthat occurs shortly before the actual threshold for the exis-tence of solutions. In the talk I will describe a new methodthat is inspired by the sophisticated but non-rigorous for-malism called Survey Propagation.

Konstantinos PanagiotouMax-Planck-Institute for [email protected]

Amin Coja-OghlanUniversity of [email protected]

MS15

The Fractal Nature of the Abelian Sandpile

The Abelian sandpile is a diffusion process for configura-tions of chips on the integer lattice; a vertex with at least 4chips topples, distributing one chip to each of its neighbors.One of the most striking unexplained features of the sand-pile is that it appears to produce terminal configurationsconverging to a peculiar fractal pattern when begun fromincreasingly large stacks of chips at the origin. We willpresent a mathematical explanation for this phenomenon.

Wesley PegdenCourant [email protected]

Charles SmartMassachusetts Institute of [email protected]

Lionel LevineCornell [email protected]

MS15

Achlioptas Processes: Recent Results and NewProblems

Recent results have shown that the phase transition of alarge class of Achlioptas processes looks qualitatively simi-lar to that of the Erdos-Renyi random graph process. I willdescribe these results and pay particular attention to onetechnique involved in the proofs, combining the differen-tial equation method with Flajolet-Sedgewick singularityanalysis. I will then propose some open questions aboutAchlioptas processes related to both random graphs andrandom constraint satisfaction problems.

Will PerkinsGeorgia [email protected]

Mihyun KangTechnische Universitat GrazInstitut fur Optimierung und Diskrete [email protected]

Joel SpencerNew York [email protected]

MS16

Rational Noncrossing Partitions

For each positive rational number α/β in lowest terms wedefine a poset of noncrossing partitions NC(α/β) with theproperty that

|NC(α/β)| = 1

2α+ β

(2α+ βα

).

For the rational number α/β = n/(kn − n+ 1) we obtainthe poset of “k-divisible” noncrossing partitions of the cycle(1, 2, . . . , kn), in which the size of each block of a partitionis divisible by k. For k = 1 we obtain the “good old’NC(n) = NC(n/1).

Drew Armstrong

68 DM12 Abstracts

University of [email protected]

Nathan WilliamsUniversity of [email protected]

MS16

Separation Probabilities for Products of Permuta-tions

We study the mixing properties of permutations obtainedas product of two uniformly random permutations of fixedtypes. For instance, we give an exact formula for the prob-ability that elements 1, 2, . . . , k are in distinct cycles of therandom permutation of {1, 2, . . . , n} obtained as productof two uniformly random n-cycles.

Olivier [email protected]

Rosena DuEast China Normal [email protected]

Alejandro [email protected]

Richard StanleyMassachusetts Institute of [email protected]

MS16

Bijections for Lattice Paths Between Two Bound-aries

We prove that on the set of lattice paths with north andeast (unit) steps that lie between two boundaries B andT , the statistics ‘number of east steps shared with B’ and‘number of east steps shared with T ’ have a symmetricjoint distribution. We give an involution that switchesthese statistics, and a generalization to paths that con-tain south steps. We show that a similar result relates tothe Tutte polynomial of a matroid. Finally, we extend ourmain theorem to k-tuples of paths, and we provide con-nections to flagged semistandard Young tableaux and tok-triangulations.

Sergi ElizaldeDepartment of [email protected]

Martin [email protected]

MS16

Tableaux and Plane Partitions of Truncated Shapes

We consider a new kind of straight and shifted plane par-titions/Young tableaux — ones whose diagrams are nolonger of partition shape, but rather Young diagrams withboxes erased from their upper right ends. We find formu-las for the number of standard tableaux in certain cases,

namely a shifted staircase without the box in its upperright corner, i.e. truncated by a box, a rectangle truncatedby a staircase and a rectangle truncated by a square minusa box. The proofs involve finding the generating functionof the corresponding plane partitions using interpretationsand formulas for sums of restricted Schur functions andtheir specializations. The number of standard tableaux isthen found as a certain limit of this function.

Greta [email protected]

MS16

The Mobius Function of Generalized Subword Or-der

Let P be a poset and let P ∗ be the set of all finite lengthwords over P . Generalized subword order is the partialorder on P ∗ obtained by letting u ≤ w if and only if thereis a subword u′ of w having the same length as u such thateach element of u is less than or equal to the correspondingelement of u′ in the partial order on P . Classical subwordorder arises when P is an antichain, while letting P be achain gives an order on compositions. For any finite posetP , we give a simple formula for the Mobius function of P ∗

in terms of the Mobius function of P . This permits us torederive in an easy and uniform manner previous results ofBjorner, Sagan and Vatter, and Tomie. We are also able todetermine the homotopy type of all intervals in P ∗ for anyfinite P of rank at most 1. This is joint work with PeterMcNamara

Bruce SaganMichigan State UniversityDepartment of [email protected]

MS17

Closed 2-cell Embeddings Under Partial Duality

In 2009 Chmutov introduced a partial duality operation,using only a subset of the edges, for graph embeddings.The topological consequences of this operation have notyet been intensively investigated. One important propertyof embeddings is being closed 2-cell , when the boundary ofevery face is a cycle in the graph. We present a necessaryand sufficient condition for a closed 2-cell embedding toremain closed 2-cell after taking a partial dual.

Mark EllinghamDepartment of MathematicsVanderbilt [email protected]

Xiaoya ZhaMiddle Tennessee State [email protected]

MS17

Biembedding Designs and Minimum Genus Em-beddings

In this presentation I will discuss results establishing, forn ≡ 3 (mod 36), that Kn has a face 2-colourable, blue andgreen say, nonorientable embedding in which there are (n−1)/2 blue faces each of which have a Hamilton cycle as theirfacial walk and n(n−1)/6 green faces each of which have a

DM12 Abstracts 69

triangle as their facial walk. By adapting the constructionresults on minimum genus embeddings of Kn+Km are alsoobtained.

Tom McCourtUniversity of [email protected]

MS17

Genus Distribution of Path-like Graphs: TransferMatrix Method

The Transfer Matrix approach is taken for computing thegenus distribution of graphs whose structure is repeated ina way of a long path or a cycle. As an example we provideexplicit solutions for genus distributions of the Cartesianproduct of various small graphs with a long path or a cycle.

Bojan MoharSimon Fraser [email protected]

MS17

Polychromatic Coloring of Graphs on Surfaces

Let G be a graph on a surface. A polychromatic k-coloringof G is a vertex-k-coloring of G such that each face of Greceives all the k colors. Recently, Horev et al. proved thatevery cubic bipartite plane graph admits a polychromaticproper 4-coloring, where “4” is best possible, and the cu-bicity and bipartiteness cannot be omitted in the theorem.In my talk, we discuss a recent progress on polychromaticcolorings of graphs on surfaces.

Atsuhiro NakamotoYokohama National [email protected]

MS17

Obstructions for Embeddings of Graphs in Surfaces

Only for two surfaces, the sphere and the projective plane,the complete list of obstructions (forbidden minors) for em-bedding graphs into the surface is known. We present ourgeneral results about the obstructions of connectivity 2. Asa consequence, we obtain all obstructions for the torus ofconnectivity 2.

Bojan Mohar, Petr SkodaSimon Fraser [email protected], [email protected]

MS18

The Δ2 Conjecture for Graph Labellings with Sep-aration Conditions

In 1988 Roberts described a problem posed to him by Lan-fear concerning the efficient assignment of channels to anetwork of transmitters in the plane. To understand thisproblem, Griggs and Yeh introduced the theory of integervertex λ-labellings of a graph G. To prevent interference,labels for nearby vertices must be separated by specifiedamounts ki depending on the distance i, 1 ≤ i ≤ p. Oneseeks the minimum span of such a labelling. The p = 2case with k1 = 2 and k2 = 1 has attracted the most atten-tion, particularly the tantalizing conjecture that for such“L(2, 1)-labellings”, if G has maximum degree Δ ≥ 2, then

the minimum span is at most Δ2. It has now been provenfor all sufficiently large Δ, but remains open for small Δ,even for Δ = 3.


MS18

Backbone Coloring: Tree Backbone in PlanarGraphs

Backbone coloring is a variation of channel assignmentproblem. For a graph G and a subgraph H (called back-bone) of G, a backbone k-coloring of G over H is a propervertex coloring of G using colors from {1, 2, . . . , k}, suchthat the colors for any two adjacent vertices in H differby at least two. The backbone chromatic number of Gover H , BBC(G,H), is the smallest k of a backbone k-coloring admitted by G over H . Broesma et al. showedthat BBC(G,H) ≤ 2χ(G) − 1 holds for every G and H .This implies that if G is planar, then BBC(G,H) ≤ 7 forall H . They conjecture that if G is planar and T is a tree,then BBC(G,H) ≤ 6. We prove this conjecture when T isa tree of diameter at most 4.

Victor CamposUnversidade Federal do [email protected]

Frederic HavetINRIA Sophia-Antipolis, [email protected]

Rudini Sampaio, Ana Shirley Ferreira SilvaUniversidade Federal do [email protected], [email protected]

MS18

L(2,1,1)-Labeling Is NP-complete for Trees

An L(p1, p2, p3)-labeling of a graph G with span λ is amapping f that assigns each vertex u of G an integer la-bel 0 ≤ f(u) ≤ λ such that |f(u) − f(v)| ≥ pi wheneververtices u and v are of distance i for i ∈ {1, 2, 3}. Weshow that testing whether a given graph has an L(2, 1, 1)-labeling with some given span λ is even -complete for theclass of trees.

Petr A. GolovachDurham [email protected]

Bernard LidickyUniversity of Illinois, Urbana, [email protected]

Daniel PaulusmaDurham [email protected]

MS18

Backbone Colorings of Graphs with Large Girths

Backbone coloring is a variation of the channel assignmentproblem. For a graph G and a subgraph H (called back-bone) of G, a backbone k-coloring of G over H is a propervertex coloring of G using colors from {1, 2, . . . , k}, such

70 DM12 Abstracts

that the the colors of adjacent vertices in H differ by atleast two. The backbone chromatic number of G over H ,BBC(G,H), is the smallest k of a backbone k-coloring ofG over H . Broersma, Fomin, Golovach, and Woegingershowed that BBC(G,H) ≤ 2χ(G) − 1 holds in general.Mistkuf, Skrekovski, and Tancer proved that for any nthere exists a triangle-free graph G with a spanning treeT such that χ(G) = n and BBC(G, T ) = 2n − 1. We gen-eralize this result: For any n there exists a graph G witharbitrarily large girth and a tree T ⊆ G with χ(G) = nand BBC(G,H) = 2n − 1. Moreover, we prove that if Tis a tree with degrees bounded by a given constant, and Gis a subgraph of the square of T , then BBC(G, T ) can bedetermined in polynomial time.

Yuehua BuZhejiang Normal University, [email protected]

Daphne D. LiuCalifornia State University, Los AngelesDepartment of [email protected]

Xuding ZhuZhejiang Normal University, [email protected]

MS18

Distance Three Labellings of Graphs

Let h be a positive integer. An L(h, 1, 1)-labelling of a (fi-nite or infinite) graph is an assignment of nonnegative in-tegers (labels) to its vertices such that adjacent vertices re-ceive labels with difference at least h, and vertices distancetwo or three apart receive distinct labels. The span of sucha labelling is the difference between the maximum and min-imum labels used. Motivated by applications in frequencyassignment, the L(h, 1, 1)-labelling problem seeks for theminimum span over all L(h, 1, 1)-labellings of a graph to-gether with an optimal L(h, 1, 1)-labelling. In this talk Iwill review recent results on the L(h, 1, 1)-labelling prob-lem.

Sanmaing ZhouThe University of [email protected]

MS19

Cycle Extension Property in BIBD Block-Intersection Graphs

A cycle C in a graph G is said to be extendible if thereexists a cycle C′ such that V (C) ⊆ V (C′) and |V (C′)| =|V (C)|+1. A graph G is said to be cycle extendible if everynon-Hamiltonian cycle of G is cycle extendible. A balancedincomplete block design BIBD(v, k, λ) consists of a set ofblocks, each of which is a k-subset of a point set V of cardi-nality v, such that each pair of points occurs in precisely λof the blocks of the design. The block-intersection graph ofa design D is the graph having the block set of D as its ver-tex set, and in which two vertices are adjacent if and onlyif their corresponding blocks have non-empty intersection.We show that for integers k ≥ 2 and λ ≥ 1, the block-intersection graph of a BIBD(v, k, λ) is cycle extendable.This is joint work with David Pike.

Atif AbueidaThe University of Dayton

[email protected]

David PikeMemorial University of [email protected]

MS19

Schroder Quasigroups and Related CombinatorialDesigns

Schroder quasigroups have been studied quite exten-sively over the years. Apart from corresponding toself-orthogonal Latin squares with the Weisner property,Schroder quasigroups are known to be associated withother combinatorial configurations such as orthogonal ar-rays with interesting conjugate invariant properties, a classof edge-colored designs with block size 4, and triple tour-naments. The purpose of this talk is to survey known ex-istence results for various aspects of these associations andto mention some open problems.

Frank BennettMount Saint Vincent [email protected]

MS19

Decompositions of Complete Graphs into Cyclesand Related Problems

In 1981, Alspach posed the problem of proving that the ob-vious necessary conditions for a decomposition of a com-plete graph into cycles of specified lengths are also suf-ficient. In this talk I will briefly outline a solution toAlspach’s problem and discuss other similar problems.

Darryn BryantUniversity of [email protected]

MS19

Friendship 3-hypergraphs

Let (X,B) be a set system in which B is a set of 3-subsets of X. Then (X,B) is a friendship 3-hypergraphif it satisfies the following property: for distinct elementsu, v, w ∈ X, there exists a unique fourth element x ∈ Xsuch that {u, v, x}, {u, w, x}, {v, w, x} ∈ B. If a friend-ship 3-hypergraph contains a element f ∈ X such that{f, u, v} ∈ B for all u, v ∈ X \ {f}, then it is called auniversal friend 3-hypergraph and the element f is calleda universal friend of the hypergraph. In this presentation,we will show that if (X,B) is a friendship 3-hypergraphwith |X| = n, then |B| ≥ �∈(\ − ∞)(\ − ∈)/��. In ad-dition, we will also show that if n ≡ 2, 4 (mod 6), thenthis bound is met if and only if (X,B) is a universal friend3-hypergraph.

Ben LiUniversity of [email protected]

MS19

Broadcast Systems

A broadcast system of order n is a decomposition of thecomplete directed graph Dn into n minimum broadcasttrees, one rooted at each node. I will show that broadcastsystems exist for all orders, and discuss some partial results

DM12 Abstracts 71

on uniform broadcast systems where all of the broadcasttrees are isomorphic.

Matthew WalshIndiana-Purdue University, Fort [email protected]

MS20

Graph Theoretical Design Strategies for DNA Self-assembly

Recent advances in DNA self-assembly have resulted innanoscale graphs: cubes, octahedrons, truncated octahe-dra, and even buckyballs, as well as ultra-fine meshes, lid-ded boxes and 2D figures. These constructs serve emergentapplications in biomolecular computing, nanoelectronics,biosensors, drug delivery systems, and organic synthesis.One construction method uses k-armed branched junctionmolecules, called tiles, whose arms are double strands ofDNA with one strand extending beyond the other, forminga ’sticky end’ at the end of the arm that can bond to anyother sticky end with complementary Watson-Crick bases.A vertex of degree k in the target graph is formed from ak-armed tile, and joined sticky ends form the edges. An-other construction method ’threads’ a single strand of DNAthrough the graphical structure and then uses short ’staple’strands for an origami folding of the DNA into the desiredgeometric realization of the graph. A third method usescircular single strands of DNA to trace the faces of a topo-logical embedding of the graph. We use graph theory todetermine optimal design strategies for biologists produc-ing these nanostructures, and conclude with a discussion ofhow the same mathematics, on the macroscale now, maybe adapted to space applications. This is joint work withGreta Pangborn, with an undergraduate research compo-nent.

Joanna Ellis-MonaghanSaint Michaels CollegeDepartment of [email protected]

MS20

Fluid Models for Self-organized Microtubule Ar-rays

In plant cells, microtubules self-organize into ordered ar-rays from an initially isotropic system as a result of theirinteractions. We develop a mean-field model which is usedto derive sufficient conditions for self-organization by con-ducting a stability analysis. Considering parameter regionsthat satisfy conditions for organization, we develop predic-tive methodologies for expected number and average lengthof microtubules over time using fluid models for micro-tubule dynamics and approximations. Results are testedusing computer simulations.

Ezgi ErenIndustrial and Systems EngineeringTexas A&M [email protected]

Natarajan GautamTexas A&M [email protected]

Ram DixitBiology DepartmentWashington University in St. Louis

[email protected]

MS20

Specification and Optimization of Synthetic Multi-cell Behaviors

The process by which a single cell develops into a multi-celled organism is complicated. A combination of internallogic, control of growth and division, and cell-cell commu-nication must coordinate the behaviors of the growing or-ganism so that it differentiates correctly in time and space,in spite of environmental perturbations, and intrinsic orextrinsic noise. Although several formalisms have been de-fined to describe development to produce CGI plants usingL-systems for example the relationship between high levelprograms and the possible biochemical implementation ofthose programs in micro-organisms is tenuous. Said differ-ently, it is unknown whether the basic mechanisms avail-able to the synthetic biologist are sufficient to implementthe pattern formation algorithms defined at higher levels.Here, we introduce a formal specification and programminglanguage, called gro, that allows the programmer to writedistributed algorithms at a variety of different levels of ab-straction. For example, the programmer may specify thata cell changes from one state to another, of that cell stateis defined at the level of a multi-stable gene network withall of the low-copy number noise that would accompanyit. The programs can be simulated in an environment thatmodels the basic geometry of micro-colony growth and di-vision, as one might see under a microscope, and also mod-els the production, degradation, and diffusion of signalingmolecules. In this talk, we illustrate gro with a variety ofexamples; focus on the example of symmetry-breaking inparticular; and discuss a notion abstraction for gro pro-grams using Wasserstein pseudometrics. The result is atool that may someday allow the molecular programmer tospecify a developmental process at a high level, and refineit, step-by-step, into a low level description of how an im-plementation would work, and finally to formally comparethe specification to an actual experimental implementa-tion.

Eric KlavinsDepartment of Electrical EngineeringUniversity of [email protected]

MS20

Inferring Physical Parameters and Assembly Path-ways from Indirect Measures of Viral Self-assembly

Virus capsids have inspired many theoretical studies of as-sembly mechanisms and pathways accessible to a simple setof self-assembling components. It has generally proven im-possible, though, to determine where any particular virusfalls in this universe of possibilities. We have combined dis-crete assembly models with continuous optimization overrate parameters to infer assembly kinetics and pathwaysfrom bulk measures of assembly progress. Application toreal viruses suggests a diversity of mechanisms in use innature.

Russell SchwartzBiological Sciences and Computer Science DepartmentsCarnegie Mellon [email protected]

Lu XieJoint Ph.D. Program in Computational Biology

72 DM12 Abstracts

Carnegie Mellon [email protected]

Gregory SmithDepartment of BIological SciencesCarnegie Mellon [email protected]

Xian FengDepartment of Biological SciencesCarnegie Mellon [email protected]

MS20

Modelling the Co-operative Roles of Genomic RNAduring Virus Assembly

We demonstrate that the genomes of ssRNA viruses im-pact on the assembly of the viral protein containers thatpackage them. For a test virus, bacteriophage MS2, weshow that its genomic sequence has evolved specific pat-terns that play important functional roles in the assemblyprocess. We quantify these via kinetic modelling and acomputation of the assembly pathways and, in combina-tion with graph theory, predict unprecedented details con-cerning the organisation of the packaged genome.

Reidun TwarockDepartments of Mathematics and BiologyUniversity of [email protected]

Eric Dykeman, Nick GraysonDepartment of BiologyUniversity of [email protected], [email protected]

Peter StockleyAstbury Centre for Structural Molecular BiologyUniversity of [email protected]

MS21

Egervary LPs and Fractional Vertex Packing

We treat relations of Egervary (i.e. Hungarian) type algo-rithms to various early results on fractional vertex packingsand coverings by Nemhauser and Trotter, Picard and Quer-anne, Bourjolly and Pulleyblank, Lovasz, Korach, Dem-ing, Sterboul, and Kayll, including that, for any vertexweighted graph G, every maximal set of vertices with 0sand 1s, rather than halves, in an optimum weight frac-tional vertex packing is the same. It can be extended to anoptimum integer vertex packing of G. It provides an easyway to test whether G is K-E since, with all weights one,some optimum fractional vertex packing has no fractions ifand only if G is K-E.

Jack [email protected]

MS21

Konig-Egervary Graphs: Introduction and aWarm-up Result

K-E graphs are those for which the maximum size of amatching coincides with the minimum size of a covering

(bipartite graphs being a basic example). Edmonds (1965)characterized the perfect matching polytope PM(G) of agraph G = (V,E) as the set of nonnegative x ∈ �E sat-isfying two families of constraints: ‘vertex saturation’ and‘blossom’. When the blossom constraints are redundant fordetermining PM(G), the graph is non-Edmonds. After dis-cussing some early results on K-E graphs, we’ll show thatthese graphs are non-Edmonds.

Mark KayllUniversity of [email protected]

MS21

Fractional Independence Number and Konig-Egervary Graphs

The fractional independence number αf of a graph is themaximum value of

∑w(vi), where w(vi) ∈ [0, 1], for each

vertex vi of the graph, and w(vi) + w(vj) ≤ 1 for eachpair {vi, vj} of adjacent vertices in the graph. This is thelinear programming relaxation of the integer programmingformulation for the independence number α of the graph.αf is an upper bound for α and can be computed efficiently.We show that α = αf for a graph if, and only if, thegraph is Konig-Egervary, and discuss connections to thework of Nemhauser, Trotter, Bourjolly, Pulleyblank, andthe theory of critical independent sets.

Craig E. LarsonVirginia Commonwealth [email protected]

MS21

Independent Sets in almost Konig-EgervaryGraphs

Let α(G) denote the cardinality of a maximum indepen-dent set and μ(G) the size of a maximum matching inG = (V,E). If α(G) + μ(G) = |V | − 1, then G is analmost Konig-Egervary graph. Let core(G) (corona(G)) bethe intersection (union, respectively) of all maximum inde-pendent sets, and ker (G) be the intersection of all criticalindependent sets. In this talk we prove that if G is an al-most Konig-Egervary unicyclic graph, then: (i) ker (G) =core(G) and (ii) |corona(G)|+ |core(G)| = 2α (G) + 1.

Vadim E. LevitAriel University Center of Samaria and Holon Inst. ofTech.Dept. of Computer Science & [email protected]

Eugen MandrescuDepartment of Computer ScienceHolon Institute of Technologyeugen [email protected]

MS22

Phase Transition in Random Integer Programs

We consider integer programs on random polytopes in Rn

with m facets whose normal vectors are chosen indepen-dently from any spherically symmetric distribution. Weshow a phase transition phenomenon: a transition frominteger infeasibility to integer feasibility happens withina constant factor increase in the radius of the largest in-scribed ball. Our main tools are: a new connection between

DM12 Abstracts 73

integer programming and matrix discrepancy, a bound onthe discrepancy of random Gaussian matrices and Lovett-Meka’s algorithm for finding low discrepancy solutions.

Karthekeyan ChandrasekaranGeorgia Institute of [email protected]

MS22

The Game Chromatic Number of Sparse RandomGraphs

Given a graph G and an integer k, two players properlycolor the vertices of G using k colors. The first playerwins iff when the game ends all the vertices are colored.The game chromatic number χg(G) is the minimum k forwhich the first player has a winning strategy. We presentresults regarding the asymptotic behavior of this parameterfor random graphs with constant average degree and forrandom regular graphs.

Simi HaberTel Aviv [email protected]

Alan Frieze, Mikhail LavrovCarnegie Mellon [email protected], [email protected]

MS22

Connectivity and Giant Components in StochasticKronecker Graphs

A stochastic Kronecker graph is a random graph modelwherein each vertex is represented by a string in an alpha-bet Γ, and the probability that two vertices are adjacent isdetermined by comparing their corresponding strings. Weuse matrix concentration inequalities to develop conditionsunder which a stochastic Kronecker graph is asymptoticallyconnected, and under which it has a giant component.

Mary RadcliffeUC San [email protected]

MS22

The Power and Weakness of Two Choices: Unbal-anced Allocations

The ”power of two choices” (generating several randomoptions and choosing among them) has been used success-fully in load balancing algorithms, queuing theory, andAchlioptas processes. Here an allocation algorithm for gen-erating an unbalanced distribution using this technique ispresented and analyzed. This type of algorithm has con-nections to physics, biology, and economics, and aroseoriginally from a real-world cost-minimization question.Its analysis uses ideas from differential equations, randomwalks, and some new concepts.

Amanda [email protected]

MS23

Symmetric chain decomposition for necklace posets

and relation to cyclic crystals

The quotient of a Boolean algebra by a cyclic group isproven to have a symmetric chain decomposition. Thisgeneralizes earlier work of Griggs, Killian and Savage onthe case of prime order, giving an explicit construction forany order, prime or composite. The combinatorial mapspecifying how to proceed downward in a symmetric chainis shown to be a natural cyclic analogue of Kashiwara’s sl2lowering operator in the theory of crystal bases. This isjoint work with Anne Schilling.

Patricia L. HershNorth Carolina State [email protected]

MS23

Volume Bounds for Shadow Covering

Suppose that K and L are compact convex subsets of Eu-clidean space such that, for every direction u, the orthogo-nal projection of L onto the subspace u⊥ contains a trans-late of the corresponding projection of the body K. Inspite of these covering conditions, it is possible for K tohave greater volume than L. The question then turns to

bounds: How large can the volume ratio V (K)V (L)

be? And

what other valid comparisons can be made? While thesequestions are posed in the setting of convex geometric anal-ysis, a fundamental role in their resolution is played byHelly’s Theorem.

Dan KlainUMass Lowelldaniel [email protected]

Christina ChenNewton North High School-

Tanya [email protected]

MS23

Perturbation of Transportation Polytopes

We introduce a perturbation method that can be used toreduce the problem of finding the multivariate generatingfunction (MGF) of a non-simple cone to computing theMGF of simple cones. We then give a universal perturba-tion that works for any transportation polytope. We applythis perturbation to the family of central transportationpolytopes of order kn × n, and obtain formulas for theMGFs of the feasible cones of vertices of the polytope andthe MGF of the polytope. The formulas we obtain are enu-merated by combinatorial objects. A special case of theformulas recovers the results on Birkhoff polytopes givenby the author and De Loera and Yoshida.

Fu LiuUC [email protected]

MS23

Monotone Hurwitz Numbers: Polynomiality andExplicit Forms

Monotone Hurwitz numbers are a desymmetrized version

74 DM12 Abstracts

of the classical Hurwitz numbers from enumerative alge-braic geometry. They appeared recently in the context ofrandom matrix theory. In this talk I will focus on the al-gebraic and combinatorial structure of monotone Hurwitznumbers, which is to a surprising extent similar to the clas-sical case. In particular, I will discuss polynomiality andToda equations for monotone Hurwitz numebers, as wellas low-genus formulas. This is joint work with I. Gouldenand M. Guay-Paquet (Waterloo).

Jonathan I. [email protected]

MS23

Orientations, Semiorders, Arrangements, andParking Functions

It is known that the Pak-Stanley labeling of the Shi hy-perplane arrangement provides a bijection between the re-gions of the arrangement and parking functions. For anygraph G, we define the G-semiorder arrangement and showthat the Pak-Stanley labeling of its regions produces all G-parking functions.

David PerkinsonReed [email protected]

MS24

Distances Between Evolutionary Trees

Phylogenetic (leaf-labelled) trees are used in computationalbiology to show the evolutionary relationships within a col-lection of species. A common way to measure the dissim-ilarity between phylogenetic trees with identical leaf setsis the minimum number of certain tree rearrangement op-erations that is necessary to transform one tree into theother. In other words, we use the graph distance of thegraph where the vertices are all phylogenetic trees withgiven leaf set and two trees are adjacent if they are a singlerearrangement operation apart. I will summarize recent re-sults on the tree bisection and reconnection (tbr) and thesubtree prune and regraft (spr) distances.

Stefan [email protected]

MS24

On t-Path Closed Graphs

A t-path closed is a graph with girth t+1 such that for eachpair of distinct vertices there is a path of length t connect-ing them. In this talk we will give several constructionsof such graphs. Also we will give relations with cages andMoore graphs.

Jacobus KoolenKorea Advanced Institute of Science and TechnologyDept Mathematics,POSTECH,[email protected]

Hwang Rae LeeDept [email protected]

Stefan [email protected]

MS24

Switchings, Extensions, and Reductions in CentralDigraphs

A directed graph is called central if, for each ordered pairu, v of (not necessarily distinct) vertices, there is a uniquevertex z such that the digraph contains the arcs uz, zv.Equivalently, the adjacency matrix A satisfies the matrixequation A2 = J , where J is the matrix with a 1 in eachentry. It has been conjectured that every central directedgraph can be obtained from a standard example by a se-quence of simple operations called switchings, and also thatit can be obtained from a smaller one by an extension. Wedisprove these conjectures and present a general extensionresult which, in particular, shows that each counterexam-ple extends to an infinite family of central digraphs.

Andre KundgenCalifornia State University, San [email protected]

Gregor LeanderTechincal University of [email protected]

Carsten ThomassenTechnical University of [email protected]

MS24

Greedy Trees and the Extremal Distances

We show a “universal property’ of the greedy tree with agiven degree sequence, namely that the number of pairsof vertices whose distance is at most k is maximized bythe greedy tree for all k. This rather strong assertion im-mediately implies, and is equivalent to, the minimality ofthe greedy trees with respect to graph invariants of theform Wf (T ) =

∑{u,v}⊆V (T )

f(d(u, v)) for any nonnega-

tive, nondecreasing function f . With different choices of f ,one directly solves the minimization problems of distance-based graph invariants including the classical Wiener in-dex, the Hyper-Wiener index and the generalized Wienerindex.

Nina SchmuckTechnische Universit”at [email protected]

Stephan WagnerStellenbosch [email protected]

Hua WangGeorgia Southern University, [email protected]

MS24

Walks and homomorphisms of digraphs

In this talk we try to discuss some connections between

DM12 Abstracts 75

symbolic dynamics and digraph homomorphisms and thenapply them to some problems on walks in digraphs posedby Delorme, Tvrdik, Harbane and Heydemann et al.

Yaokun WuShanghai Chiao Tong Univ.,Shanghai, [email protected]

Yaokun WuShanghai Jiao Tong UniversityShanghai, [email protected]

MS25



Andreas BrandstadtFB Informatik, University of Rostock, [email protected]

MS25

Convexity and Graph Classes

We survey graph classes that arise from convexity in graphswith emphasis on convex geometries.

Ortrud R. OellermannThe University of [email protected]

MS25

Atomic Structure, Hyperbolicity, and Recognitionof AT-free Graphs with no Induced 4-cycles

An atom of a graph is a maximal induced subgraph withno clique cutset. We characterize atoms of C4-free AT-freegraphs. This class of graphs is of interest as it coincideswith 1-hyperbolic AT-free graphs while all other AT-freegraphs are 2-hyperbolic. Based on the structure of atoms,we describe an efficient algorithm for the recognition of theclass.

Derek CorneilUniversity of [email protected]

Juraj StachoUniversity of [email protected]

MS25

Polygon Numbers of Circle Graphs

Circle graphs and k-polygon graphs are intersection graphsof chords of a circle and chords of a convex k-sided poly-gon where each chord has its endpoints on distinct sides,respectively. Determining ψ(G), the minimum number ofsides in a polygon representation of a circle graph G, isNP-hard, whereas determining whether G is a k-polygongraph for fixed k can be done in polynomial time. We givebounds on ψ(G) when G is an arbitrary circle graph andidentify graph classes for which ψ(G) can be computed inpolynomial time.

Lorna Stewart, Richard Valenzano

University of [email protected], [email protected]

MS25

Unit Interval Graphs of Mixed Intervals

The class of unit interval graphs has a lovely characteriza-tion as those interval graphs with no induced claw K1,3.The characterization remains the same whether the inter-vals used in the intersection representation are all openintervals or all closed intervals. In recent work, Rauten-bach and Szwarcfiter characterize the broader class thatarises when both open and closed intervals of unit lengthare permitted. In this talk we consider the same problemwhen unit length mixed intervals of the form (x, x+ 1] and[x, x+ 1) are also allowed. We give a structural character-ization of this class of graphs.

Ann N. TrenkWellesley [email protected]

Alan ShuchatMathematics DepartmentWellesley [email protected]

Randy ShullDepartment of Computer ScienceWellesley [email protected]

Lee WestWellesley [email protected]

MS26

The Cost of 2-Distinguishing Cartesian Powers

A graph is said to be 2-distinguishable if there is a ver-tex labeling with two labels so that only the trivial au-tomorphism preserves the labels. Define the cost of 2-distinguishing G, denoted ρ(G), to be the minimum size ofa label class in such a labeling. A determining set is a sub-set of vertices with trivial pointwise stabilizer; its minimumsize is denoted Det(G). The main result of this talk is that

under mild hypotheses, ρ(Gk) ∈ {Det(Gk),Det(Gk) + 1}.Debra L. BoutinHamilton [email protected]

MS26

Groups That Are Transitive on All Partitions of aFinite Set

We study the question of which subgroups of Sn aretransitive on the set of all ordered partitions of the set[n] = {1, . . . , n} and all unordered partitions of [n]. Thiswork can be considered a generalization of well known workin which subgroups of Sn that are transitive on the set ofall subsets of [n] of size k were determined. As an appli-cation of our results, we determine which Johnson graphsare Cayley graphs.

Ted DobsonMississippi State [email protected]

76 DM12 Abstracts

Aleksander MalnicUniversity of [email protected]

MS26

Decomposing Hypergraphs on Finite Fields

We examine edge decompositions of complete uniform hy-pergraphs whose parts are permuted transitively by somepermutation of the vertex set. We present an algebraicmethod for constructing such a hypergraph decompositionon a finite field which is related to the Paley graph con-struction and is derived from a partition of the cosets ofthe multiplicative group of units of the field. We discussthe symmetry and other properties of the hypergraphs weobtain.

Shonda GosselinUniversity of WinnipegDepartment of Mathematics and [email protected]

MS26

Infinite Motion and Distinguishing Number 2

Given a group A acting faithfully on a set X, definethe motion (also known as minimal degree) of A, to bem(A) = mina �=1 |{x : xa �= x}|. A general principle forfinite A is that if m(A) >> |A|, then the action has dis-tinguishing number 2, that is, there is a subset Y whoseset-wise stabilizer is trivial. For infinite A and countableX, we make the Infinite Motion Conjecture: if m(A) is in-finite, then the action has distinguishing number 2. Weconsider various contexts for the conjecture in the case oflocally finite graphs.

Thomas TuckerColgate [email protected]

Wilfriied ImrichUniversity of [email protected]

Simon Smith, Mark E. WatkinsSyracuse [email protected], [email protected]

MS26

Finite Subgraphs of d-Distinguishable, Locally Fi-nite Graphs

It is known (M.E. Watkins and X. Zhou, 2007) that everyinfinite, locally finite tree T with finite distinguishing num-ber d(T ) = d0 contains a finite subtree with distinguishingnumber d0. It is not difficult to prove more generally thatif every finite subgraph Φ of an infinite, locally finite graphΓ satisfies d(Φ) ≤ d0, then d(Γ) ≤ d0 + 1. We investigateconditions subject to which this bound may be sharp.

Mark E. Watkins, Simon SmithSyracuse [email protected], [email protected]

MS27

Fast Algorithms for Phylogenetic Reconstruction

We present two recent very fast algorithms to infer phy-logenetic trees in o(n2) time when the input consists ofdata from n species. Our first algorithm uses a searchtree much like a randomly balanced binary search tree toplace each new species into its correct place. Our secondalgorithm uses nearest-neighbour search to speedily findroughly where each new taxon goes, and then uses morerobust methods to exactly place the new species correctly.

Dan BrownCheriton School of Computer ScienceUniversity of [email protected]

MS27

Gene Family Evolution by Duplication and Loss -Reconciliation and Species Tree Inference

Almost all genomes which have been studied contain genesthat are present in two or more copies. They are usu-ally identified through sequence similarity, and groupedinto a single “Gene Family’. From a functional point ofview, such grouping is not sufficient to infer a commonfunction for genes. Indeed, it is important to distinguishbetween “orthologs’ which are copies in different speciesrelated through speciation, and thus likely to have simi-lar functions, and “paralogs’, which are copies that haveevolved by duplication, and more likely to have acquirednew functions. Understanding the evolution of gene fami-lies through speciation, duplication, and loss is thus a fun-damental question in functional genomics, and also in evo-lutionary biology and phylogenomics. Reconciliation is thecommonly used method for inferring the evolutionary sce-nario for a gene family. It consists in “embedding’ inferredgene trees into a a species tree. When a species tree is notknown, a natural algorithmic problem is to infer a speciestree from a set of gene trees, such that the correspondingreconciliation minimizes the number of duplications and/orlosses. The main drawback of reconciliation is that the in-ferred evolutionary scenario is strongly dependent on theconsidered gene trees, as few misplaced leaves may lead toa completely different history. In this talk, we will clarifyseveral theoretical questions and present various algorith-mic issues related to reconciliation and species tree infer-ence. In particular, we will present a strategy for “correct-ing’ or preprocessing a gene tree prior to reconciliation orspecies tree inference, and give various complexity resultsand algorithmic solutions, in both cases of a known and anunknown species tree.

Nadia S. El-MabroukDepartement d’informatique et de rechercheoperationnelleUniversite de [email protected]

MS27

Tree Compatibility, Character Compatibility, andGraph Triangulation

We consider two classic NP-complete problems in phylo-genetic tree construction, tree compatibility and charactercompatibility. In each case, the “yes’ instances are pre-cisely those for which a specific kind of triangulation existsfor a certain graph. The relationship between triangula-tions and character compatibility has been known for over

DM12 Abstracts 77

three decades (Buneman, 1974); an analogous relationshipwith tree compatibility was identified more recently (Vakatiand Fernandez-Baca, 2011). We use these characteriza-tions to gain insight into the fundamental differences intractability between the two problems.

David F. Fernandez-BacaComputer ScienceIowa State [email protected]

Sudheer VakatiDepartment of Computer ScienceIowa State [email protected]

MS27

Aspects of Fractionation; A Fundamental Evolu-tionary Process

In the evolutionary history of virtually all eukaryoticspecies is at least one whole genome duplication (WGD).Each WGD event is followed by a a period of fractiona-tion whereby only one of the copies of almost all gene pairsis shed, sometimes on one of the two homeologous chro-mosomes, and sometimes on the other. This effectivelyscrambles the gene order on chromosomes and introducesan extreme level of noise into gene order comparisons be-tween genomes descending from the WGD and those thatdiverged before this event, and also between genomes insister lineages descending from the event radiating soonafter the WGD. In these cases, fractionation is responsiblefor more gene order disruption than rearrangements suchas inversion and translocation. We discuss three aspects offractionation:

• the time course of fractionation on scales of tens tohundreds of millions of years,

• a recurrence for the run lengths of deleted and un-deleted genes on a chromosome, and

• a model for gene family size, where a genome origi-nates from more than one successive WGD or otherpolyploidization event.

David Sankoff, Chunfang ZhengDepartment of Mathematics and StatisticsUniversity of [email protected], [email protected]

MS27

Reconstruction of Certain Phylogenetic Networksfrom the Tree-Additive Distances Between TheirLeaves

Suppose N is a rooted directed graph and each leaf corre-sponds to an extant biological species. Suppose each arcof N has a nonnegative weight and inheritance of a char-acter at a hybrid vertex occurs with a certain probabilityfrom each parent. The tree-additive distance between twoleaves is the expected value of their distances in the dis-played trees. Sufficient conditions are investigated so that,given the tree-additive distances between all leaves, it ispossible to reconstruct the network.

Stephen J. WillsonIowa State [email protected]

MS28

Quotients of the Boolean Lattice by Wreath Prod-ucts

Rod Canfield has asked if all quotients of a Boolean lat-tice Bn induced by subgroups of its automorphism group,essentially Sn, are symmetric chain orders. K. Jordan ver-ified this in the special case of a subgroup generated byan n-cycle, answering a question raised earlier by Griggs,Killian and Savage. In earlier work we found an espe-cially straightforward proof of Jordan’s result based on theGreene-Kleitman symmetric chain decomposition. Hershand Schilling, and Dhand have presented other proofs ofthe same result, and the latter has the substantial general-ization that Pn/Zn is an SCO whenever P is an SCO. Herewe continue in the same vein as our earlier work, describingcircumstances under which subgroups G of the symmetricgroup which are wreath products are guaranteed to inducequotients that are symmetric chain orders. This is done bycarefully winnowing Greene-Kleitman SCDs.

Dwight DuffusEmory UniversityMathematics and Computer Science [email protected]

Kyle ThayerEmory [email protected]

MS28

Poset-free Families of Sets

Given a finite poset P , we consider the largest size La(n, P )of a family F of subsets of [n] := {1, . . . , n} that containsno (weak) subposet P . For fixed P it can be very difficultto determine La(n, P ), even asymptotically. We continue

to believe that π(P ) := limn→∞ La(n, P )/(

n�n/2�

)exists for

general posets P , and, moreover, it is an integer that wecan predict. The existence of the limit remains open, forinstance, for the four-element diamond poset and the six-element crown. However, we are developing methods thatallow us to solve the asymptotic problem, and sometimeseven the exact problem for all n, for increasingly manyposets P . These methods involve studying the Lubell func-tion of a family F , which is the average number of times arandom full chain meets F .


Wei-Tian LiDepartment of MathematicsUniveristy of South [email protected]

Linyuan LuUniversity of South [email protected]

MS28

Chopping Celery and the Lattice of Integer Parti-tions

The chop vector of a set S of celery sticks of positive integerlengths is the infinite vector vS = (v1, v2, v3, . . .), whereeach vw is the minimum number of cuts needed to chop S

78 DM12 Abstracts

into unit pieces, using a knife that can cut up to w sticks ata time. In this talk we see a connection (found jointly withThao Do) between the set of chop vectors and the latticeof integer partitions.

Bill Sands, Thao DoUniversity of [email protected], [email protected]

MS28

Diamond-Free Collections of Subspaces

Let V be a finite-dimensional vector space over a finitefield, and let A, B, C, and D be subspaces of V . We saythat {A,B,C,D} is a diamond if A is a subspace of bothB and C which in turn are subspaces of D. Inspired by therecent work on the Boolean Lattices, we ask: What is thesize of the largest collection of diamond-free subspaces ofV ? The corresponding question in the Boolean Lattices hasnot been completely settled and neither is the question inthe Linear Lattices. We report on partial progress. Jointwork with Ghassan Sarkis and the undergraduate mathresearch circle at Pomona College.

Shahriar ShahriariPomona [email protected]

MS28

Linear Discrepancy of Partially Ordered Sets

The linear discrepancy of a P , denoted ld(P ), is the min-imum over all linear extensions of the maximum differ-ence between the positions of two incomparable elements.Tanenbaum, Trenk, and Fishburn conjectured that alwaysld(P ) ≤ �(3r − 1)/2�, where r is the maximum number ofelements incomparable to any single element. We disprovethis conjecture by showing that appropriately randomizedbipartite posets have linear discrepancy asymptotic to thetrivial upper bound, 2r − 1. On the other hand, we showthat the conjecture does hold for posets having width 2.

Jeong Ok ChoiGIST CollegeGwangju Institute of Science and [email protected]

Kevin MilansUniversity of South [email protected]

Douglas B. WestUniversity of Illinois, UrbanaDepartment of [email protected]

MS29

Roots of Combinatorial Polynomials

Polynomials arise in a variety of ways in combinatorics,from applications like graph colourings and network relia-bility, to theoretical investigations of sequences related toindependent sets of graphs and open sets of finite topolo-gies. We discuss some recent results about the roots ofsuch polynomials, both in terms of location and nature, aswell as some connections to fractals.

Jason Brown

Dalhousie University, CanadaDept of Maths, Stats, & [email protected]

MS29

Graph Colouring and the Topological PenrosePolynomial

The Penrose polynomial of a plane graph first appearedimplicitly in Roger Penroses 1971 work on diagrammatictensors, but was discovered to have a number of remarkablegraph theoretical properties, particularly with respect tograph colouring. We now extend the Penrose polynomial,originally defined only for plane graphs, to graphs embed-ded in arbitrary surfaces. This leads to new identities andrelations for the Penrose polynomial which cannot be re-alized within the class of plane graphs. In particular, byexploiting connections with the transition polynomial andthe ribbon group action, we find a deletion-contraction-type relation for the Penrose polynomial and its relationto the ribbon graph polynomial of Bollobas and Riordan.We relate the Penrose polynomial of an orientable checker-board colourable graph to the circuit partition polynomialof its medial graph and use this to find new combinatorialinterpretations of the Penrose polynomial. We also showthat the Penrose polynomial of a plane graph G can be ex-pressed as a sum of chromatic polynomials of twisted dualsof G. This allows us to obtain a new reformulation of theFour Colour Theorem.

Joanna Ellis-MonaghanSaint Michaels CollegeDepartment of [email protected]

Iain MoffattUniversity of South [email protected]

MS29

Matroids, Greedoids and the Tutte Polynomial

We present some conjectures and counterexamples relatedto the greedoid generalization of the Tutte polynomial ofa matroid. In particular, we give simple examples to showthe Tutte polynomial does not distinguish greedoids. Wealso study the affine relations satisfied by the coefficientsof the Tutte polynomial in this more general setting.

Gary GordonLafayette CollegeDepartment of [email protected]

MS29

Some Recent Results on Chromatic and TuttePolynomials for Families of Graphs

We discuss some recent results on chromatic and Tuttepolynomials (equiv. to Potts model partition functionsin statistical physics) and some generalizations for variousfamilies of graphs. These include weighted-set chromaticand Tutte polynomials, lower bounds on Potts ground-state entropy, and chromatic polynomials for one- andmulti-parameter families of planar triangulation graphs.References include S.-C.Chang and R. Shrock, J. Phys. A(JPA) 42, 385004 (2009); J. Stat. Phys. (JSP) 138, 496(2010); R. Shrock and Y. Xu, JSP 139, 27 (2010); JSP

DM12 Abstracts 79

141, 909 (2010); Phys. Rev. E 81, 031134 (2010); JPA45, 055212 (2012); arXiv:1201.4200; R. Shrock, JPA 44,145002 (2011).

Robert ShrockSUNY at Stony BrookC. N. Yang Institute for Theoretical [email protected]

MS29

Peck orders and LYM orders


David WagnerUniversity of Waterloo, CanadaDept. of Combin. and [email protected]

MS30

Edge-coloring 8-regular Planar Graphs

In 1974 Seymour made the following conjecture:

Let G be a k-regular planar graph, such that if X is an oddset of vertices of G, then at least k edges of G have oneend in X and the other in V (G) \ X. Then G is k-edgecolorable.

For k = 3 this is equivalent to the Four Color Theorem.The cases of k = 4, 5 were solved by Guenin, the case ofk = 6 by Dvorak, Kawarabayashi and Kral, and the caseof k = 7 by Edwards and Kawarabayashi. In joint workwith Edwards and Seymour, we now seem to have a prooffor the case k = 8, and that is the topic of this talk.

Maria [email protected]

Katherine Edwards, Paul SeymourPrinceton [email protected], [email protected]

MS30

Graph Stirling Numbers

For a graph G and an integer k, the graph Stirling numberS(G, k) is the number of partitions of V (G) into k non-empty independent sets. Equivalently it is the number ofproper colourings of G using exactly k colours, with twocolourings identified if they differ only on the names ofthe colours. If G is the empty graph on n vertices thenS(G, k) is just S(n, k), the familiar Stirling number of thesecond kind. In this talk, we look at the Stirling numbersof families and forests and cycles. Among other results, wegeneralize Harper’s theorem on the asymptotic normalityof the ordinary Stirling numbers.

David J. Galvin, Do Trong ThanhUniversity of Notre [email protected], [email protected]

MS30

Recent Results on Grundy number

A k-coloring of a graph G = (V,E) is a function f : V →

{1, . . . , k} such that f(v) �= f(w) for all vw ∈ E. TheGrundy number of a graph is the largest integer k suchthat there is a k-coloring f of G such that if 1 ≤ i < j ≤ kand f(v) = j then there exist w ∈ N(v) such that f(w) = i.It is well known that the Grundy number of G is equal tothe worst case performance of First-Fit on G. I will discussmany recent results concerning Grundy number.

H. A. KiersteadMathematics DepartmentArizona State [email protected]

MS30

Bounding the Fractional Chromatic Number of KΔ-free Graphs

In this talk I will discuss recent work in bounding the dif-ference between the maximum degree and the fractionalchromatic number in graphs for which the answer is notobvious, i.e. for graphs whose clique number is less thanthe maximum degree. This work has interesting ties to theBorodin-Kostochka conjecture and Reed’s conjecture.

Andrew D. KingColumbia [email protected]

Linyuan Lu, Xing PengUniversity of South [email protected], [email protected]

MS30

Multicolor and Directed Edit Distance

For the edit distance problem, simple graphs on the samevertex set are elements in a metric space defined by the nor-malized edit distance. The primary question is the maxi-mum distance of a graph from some fixed hereditary prop-erty.

In this talk, we generalize this concept to other kinds ofgraphs. We consider multicolorings of the edges of com-plete graphs as well as simple directed graphs. The samequestion – the maximum distance of an element from ahereditary property – is considered in these settings andwe provide bounds for this quantity.

Ryan R. Martin, Maria AxenovichIowa State [email protected], [email protected]

MS31

Two Extensions of Ramsey’s Theorem

Ramsey’s theorem states that every 2-colouring of theedges of the complete graph on {1, 2, . . . , n} contains amonochromatic clique of order roughly log n. We provenew bounds for two extensions of Ramsey’s theorem,each demanding extra structure within the monochromaticclique. In so doing, we answer a question of Erdos and im-prove upon results of Rodl and Shelah.

David ConlonUniversity of [email protected]

Jacob Fox

80 DM12 Abstracts

[email protected]

Benny SudakovDepartment of [email protected]

MS31

On Restricted Ramsey Numbers

A classical Ramsey theorem states that in any 2-coloringof the edges of a sufficiently large complete graph, one willalways find a monochromatic complete subgraph. In 1970,Folkman extended this result showing that for any graphG there exists a graph H with the same clique numberas G such that any 2-coloring of the edges of H yields amonochromatic copy ofG. In this talk, we present some oldand recent developments concerning Folkman-type resultsfor vertex colorings.

Andrzej DudekDepartment of MathematicsWestern Michigan [email protected]

MS31

Ramsey Problems on Non-Complete Graphs

A typical question in Ramsey Theory is, for given graphsG and H , to determine the Ramsey number R(G,H): thesmallest integer N such that, in every colouring of the edgesof the complete graph KN on N vertices with red and blue,we either find a red copy of G or a blue copy of H . Onenaturally wonders whether the complete graph KN in theabove definition can be replaced by some other graph Fon N vertices. It would be naive to think we can take anygraph on N vertices, therefore, we put some restrictionson such a graph F . In this we shall discuss the followingproblem: For given graphsG andH , is there a constant 0 <c < 1 such that if a graph F on N = R(G,H) vertices hasminimum degree at least (1− c)N , then in every colouringof the edges of F with red and blue, we either find a redcopy of G or a blue copy of H . It is not even clear whysuch c should even exist. We determine the maximum cwhen G and H are cycles or paths, and show that c existsfor a rather large class of pairs (G,H).

Jozef SkokanLondon School of [email protected]

MS31



TBA TBAIBM CorporationTBA

MS31

Diagonal Forms for Incidence Matrices and Zero-Sum (mod 2) Ramsey Theory

Let H be a t-uniform hypergraph on k vertices, with ai ≥ 0denotes the multiplicity of the i-th edge, 1 ≤ i ≤ ( k

t). Let

h = (a1, · · · , a( kt))�, and Nt(H) the matrix whose columns

are the images of h under the symmetric group Sk. Wedetermine a diagonal form (Smith normal form) of Nt(H)for a very general class of H . Motivated by Y. Caro’sresults, we show that if H is simple, the zero-sum Ramseynumber (mod 2) of H is almost surely k as k→∞.

Wing Hong Tony Wong, Richard WilsonCalifornia Institute of [email protected], [email protected]

MS32

Non-extendible Latin Cuboids

There is a celebrated result due to Marshall Hall that everylatin rectangle is completable to a latin square. However,the equivalent statement in higher dimensions is not true.A 3-dimensional array has lines of cells in three directions,called rows, columns and stacks. An n×n×k latin cuboid isa 3-dimensional array containing n different symbols posi-tioned so that every symbol occurs exactly once in each rowand column and at most once in each stack. An n× n× nlatin cuboid is a latin cube of order n. An n× n× k latincuboid is extendible if it is contained in some n×n×(k+1)latin cuboid and it is completable if it is contained in somelatin cube of order n. In this talk I will present some re-sults on non-extentible and non-completable n×n×k latincuboids where k is approximately half of n.

Darryn BryantUniversity of [email protected]

Nicholas CavenaghUniversity of [email protected]

Barbara MaenhautUniversity of [email protected]

Kyle PulaUniversity of [email protected]

Ian WanlessMonash [email protected]

MS32

Block Colourings of Designs Revisited

Block colourings of designs, both classical and its variants,and also the so-called specialized block colourings, offer anextensive variety of interesting problems. We will discusssome old and new concepts, results and open problems incolourings of block designs, mostly for Steiner triple sys-tems and Steiner systems S(2, 4, v).


DM12 Abstracts 81

MS32

Skolem and Rosa Rectangles and Related Designs

A Skolem sequence of order n is a sequence S =(s1, s2, . . . , s2n) of 2n integers such that (1) for all k =1, . . . , n there are exactly two terms si, sj such thatsi = sj = k, and (2) if si = sj = k and i < j , thenj − i = k. Two Skolem sequences S, S′ are disjoint ifsi = sj = k = s′t = s′u implies that {i, j} �= {t, u}, forall k = 1, 2, . . . , n. For example, the two Skolem sequencesof order four 1, 1, 4, 2, 3, 2, 4, 3 and 2, 3, 2, 4, 3, 1, 1, 4 are dis-joint. A set ofm pairwise disjoint Skolem sequences forms aSkolem rectangle of strength m. The above sequences thenform a Skolem rectangle of strength two: 1,1,4,2,3,2,4,32,3,2,4,3,1,1,4. We introduce several new and powerful con-structions for Skolem and Rosa rectangles then we applythem to generate simple cyclic triple systems and cyclicdisjoint triple systems.

Nabil ShalabyMemorial University of [email protected]

Vaclav LinekUniversity of [email protected]

MS32

Cyclic Block Designs with Block Size 3 fromSkolem-Type Sequences

A Skolem-type sequence is a sequence (s1, . . . , st) of posi-tive integers i ∈ D such that for each i ∈ D there is exactlyone j ∈ {1, . . . , t− i} such that sj = sj+i = i. Positions inthe sequence not occupied by integers i ∈ D contain nullelements. In 1939, Peltesohn solved the existence problemfor cyclic Steiner triple systems for v ≡ 1, 3(mod 6), v �= 9.Using the same technique in 1981, Colbourn and Colbournextended the solution to all admissible λ > 1. It is knownthat Skolem-type sequences may be used to construct cyclicSteiner triple systems as well as cyclic triple systems withλ = 2. The main result of this talk is an extension of formerresults onto cyclic triple systems with λ > 2. In additionwe introduce a new kind of Skolem-type sequence. This isjoint work with my supervisor, Dr. Nabil Shalaby.

Daniela Silvesan, Nabil ShalabyMemorial University of [email protected], [email protected]

MS33

Combinatorics of Splice Graphcs

Splice graphs compactly represent the possible isoforms ofa gene which undergoes multple splicing. I will talk aboutthe combinatorics of splice graphs and how this relates tothe problem of inferring them from RNA-Seq data.

Dustin CartwrightYale [email protected]

MS33

Bayesian Centroid Estimation for Genome-WideAssociation Studies

Genome-Wide Association Studies (GWAS) attempt toidentify a (possibly small) subset of single nucleotide poly-

morphisms (SNPs) from a large number of measured can-didates that are associated with a specific observable trait.Identification of associated genetic variants is particularlyhard due to very large genotype sizes in comparison tocase-control group sizes. We formally frame this problemas Bayesian variable selection in a logistic regression modelwith spike-and-slab priors in the coefficients. To help over-come the curse of dimensionality, we set a hierarchicalmodel that groups SNPs according to gene proximity andfurther explore a co-dependency structure between SNPsand genes by adopting an Ising hyper-prior on the spaceof possible SNP associations to trait. We introduce andderive centroid and graph centroid estimators and contrastthem to the classical MAP estimators in this setup. Weillustrate this approach with a toy example and a largersimulated dataset based on HapMap. Finally, we offer afew concluding remarks and directions for future work.

Luis CarvalhoDept. of Mathematics and StatisticsBoston [email protected]

MS33

Encoding Memories in Neuronal Networks

Various mechanisms for encoding memories in networks ofneurons have been studied in the past few decades, mostnotably since Hopfield’s famous papers in the early 1980s.Given a prescribed list of memory patterns, however, it isstill an unsolved problem how to arrange connections be-tween neurons in a recurrent network such that these - andonly these - memories are encoded via stable fixed pointsof the dynamics. I will discuss some recent progress on thisproblem, which exploits connections to discrete geometry.

Carina CurtoDepartment of MathematicsUniversity of [email protected]

MS33

Assembling Helices in RNA Junctions by Using 3DGraphs

The structure of RNA junctions is important to under-stand its function. A novel 3D tree graph approach is in-troduced to predict the geometrical helical arrangementsof junctions in space. Tree graphs are implemented torepresent junctions, and biological assumptions combinedwith graph combinatorics are used to study the geometri-cal/topological configurations of native-like junctions. Fora given target junction, statistical analysis is then used topredict the tree graph that best resemble its RNA nativestructure.

Christian LaingDepartments of Biology, Mathematics and ComputerScienceWilkes [email protected]

MS33

Asymptotic Distribution of Substructures in aStochastic Context-free Grammar Model of RNAFolding

Some recent methods for predicting RNA secondarystructures are based on stochastic context-free grammars

82 DM12 Abstracts

(SCFGs). We analyze one of the most notable SCFGswhich is used in the prediction program Pfold. In par-ticular, we show that the distribution of base pairs, helicesand various types of loops in RNA secondary structuresgenerated by this SCFG is asymptotically Gaussian, for ageneric choice of the grammar probabilities. Our proofsare based on singularity analysis of probability generatingfunctions.

Svetlana PoznanovikjSchoolg of MathematicsGeorgia Institute of [email protected]


MS34

Improving Christofides’ Algorithm for the Metrics-t Path Traveling Salesman Problem

The Metric s-t Path Traveling Salesman Problem (TSP)asks for a minimum-cost Hamiltonian path with specifiedstart and end points in a finite metric space. Christofides’3/2-approximation algorithm for the Metric TSP wasshown by Hoogeveen in 1993 to be a 5/3-approximation forthe Metric s-t Path TSP. We present a polynomial-time φ-approximation algorithm for the s-t Path TSP, where φ isthe golden ratio, the first improvement upon Christofides’algorithm in general metrics.

Hyung-Chan An, Robert Kleinberg, David ShmoysCornell [email protected], [email protected],[email protected]

MS34

Electrical Flows, Laplacian Systems, and FasterApproximation of Maximum Flow in UndirectedGraphs

In this talk, I’ll describe a new technique for approximat-ing the maximum flow in capacitated, undirected graphs.I’ll then use this technique to develop the asymptoticallyfastest-known algorithm for solving this problem. Our ap-proach is based on treating the graph as a network of re-sistors and solving a sequence of electrical flow problemswith varying resistances on the edges. Each of these maybe reduced to the solution of a system of linear equationsin a Laplacian matrix, which can be solved in nearly-lineartime.

Paul Christiano, Jonathan [email protected], [email protected]

Aleksander MadryMicrosoft Research New [email protected]

Daniel SpielmanYale [email protected]

Shang-Hua [email protected]

MS34


ABstract not available at time of publication.

Prasad RaghavendraGeorgia [email protected]

MS34

The Entropy Rounding Method in ApproximationAlgorithms

Let A be a matrix and x be a fractional vector, say anLP solution to some discrete optimization problem. Wegive a new randomized rounding procedure for obtainingan integral vector y such that Ay ≈ Ay, provided that Ahas bounded Δ-approximate entropy. This property meansthat for uniformly chosen random signs χ(j) ∈ {±1} onany subset of the columns, the outcome Aχ can be ap-proximately described using at most m

5bits in expectation

(with m being the number of selected columns). To achievethis result, we modify well-known techniques from the fieldof discrepancy theory, especially we rely on Beck’s entropymethod. We demonstrate our procedure by providing anOPT +O(log2 OPT ) approximation for Bin Packing WithRejection.

Thomas RothvossMassachusetts Institute of TechnologyDepartment of [email protected]

MS34

Semidefinite Programming Hierarchies and theUnique Games Conjecture

We survey recent results about semidefinite programming(SDP) hierarchies in the context of the Unique Games Con-jecture. This conjecture has emerged as a unifying ap-proach towards settling many central open problems in thetheory of approximation algorithms. It posits the hardnessof a certain constraint satisfaction problem. We show bothupper and lower bounds on the complexity of this problemwithin restricted computational models defined by SDP hi-erarchies.

David SteurerMicrosoft Research New England, [email protected]

MS35

Some Problems and Results on the On-line ChainPartitioning of Posets

On-line chain partitioning of posets has been a widely re-searched topic since Kierstead’s seminal paper on the sub-ject in 1981. In this talk we describe a generalization ofKierstead’s auxiliary order, and we prove many powerfulproperties, in hopes of providing a linear algorithm for thechain partitioning problem.

Csaba BiroUniversity of [email protected]

Linyuan LuUniversity of South [email protected]

DM12 Abstracts 83

MS35

News About Semiantichains and Unichain Cover-ings

Saks and West ask if for any product of posets the sizeof the minimum unichain covering equals of the size of amaximum semiantichain, which would be a nice generaliza-tion of Greene–Kleitman Theorem. We found several newclasses of posets for which the conjecture is true. However,we found an example showing that in general this min-maxrelation is false. This finally disproofs 30 year old conjec-ture.

Bartlomiej E. BosekJagellion University, [email protected]

Stefan FelsnerTechnical University of BerlinInstitute for [email protected]

Kolja [email protected]

Grzegorz MateckiJagiellonian [email protected]

MS35

An Improved Bound for First-Fit on Posets With-out Two Long Incomparable Chains

It is known that the First-Fit algorithm for partitioning aposet P into chains uses relatively few chains when P doesnot have two incomparable chains each of size k. In partic-ular, if P has width w then Bosek, Krawczyk, and Szczypka(SIAM J. Discrete Math., 23(4):1992–1999, 2010) provedan upper bound of ckw2 on the number of chains used byFirst-Fit for some constant c, while Joret and Milans (Or-der, 28(3):455–464, 2011) gave one of ck2w. We prove anupper bound of the form ckw, which is best possible up tothe value of c.

Vida DujmovicCarleton [email protected]

Gwenael JoretUniversite Libre de [email protected]

David R. WoodUniversity of [email protected]

MS35

First-Fit Coloring of Interval Graphs

Let FF (k) be the maximum number of colors to be usedwhen we adopt the First-Fit algorithm to color an intervalgraph with maximum clique size k. We will discuss theestimate of FF (k) in this talk. Especially, we show that4k − 5 ≤ FF (k) ≤ 8k − 11 when k > 1, improving alittle bit the current known best record. The upper boundis obtained basically by the column construction method

invented by Pemmaraju, Raman, Varadarajan et al.

Peng LiShanghai Jiao Tong [email protected]

Yaokun WuShanghai Chiao Tong Univ.,Shanghai, [email protected]

Yaokun WuShanghai Jiao Tong UniversityShanghai, [email protected]

MS35

Semiorders and Ascent Sequences

We analyze the recent bijection of Bousquet-Melou, Claes-son, Dukes and Kitaev between interval orders and ascentsequences restricted the the case of semiorders, resulting ina new structural result on the canonical representation ofsemiorders.

Stephen J. Young, Jeff RemmelUniversity of California, San [email protected], [email protected]

MS36

Algorithms and Existential Polytime?

An EP theorem asserts the existence of something forwhich any instance is ‘easy’ to recognize. Often an EPtheorem can be proved by a polytime algorithm for findingan instance of what is asserted to exist. We discuss favoriteopen problems about the following two EP theorems: (1) Ifthere is a Hamiltonian cycle H containing edge e in graphG such that G minus the edges of H is connected, thenthere is another one H’, different from H. (2) If a surfacetriangulation S contains a set T of triangles which parti-tion the vertices of S, then S contains another such set T’,different from T.

Jack [email protected]

MS36

Some Graph Theory Problems I Would Like to SeeSolved

I will describe a number of open graph theory problemswhich arise primarily out of questions in Ramsey theory.

Ron [email protected]

MS36

Euler’s Rigidity Conjecture

In 1766, Euler conjectured that every polyhedron was rigid.Connelly finally disproved the conjecture in 1977. The in-vestigations of the conjecture and rigidity in general in-cluded many other interesting conjectures; perhaps the

84 DM12 Abstracts

most interesting has yet to be formulated.

Jack GraverMathematics DepartmentSyracuse [email protected]

MS36

Ringel and Kotzig after Fifty Years

Next year will mark the 50th anniversary of the firsttruly large international graph theory conference whichtook place in Smolenice, Czechoslovakia. Many outstand-ing problems were posed there, including the celebratedRingel’s problem on decomposing the complete graph intoisomorphic trees, and Kotzig’s problem on perfect one-factorizations of the complete graph. We will discuss somedevelopments related to these problems which remain stillwide open today.


MS37

Cops and Robbers on Geometric Graphs

In the game of cops and robbers, one robber is pursuedby a set of cops on a graph G. In each round, theseagents move between vertices along the edges of the graph.The cop number c(G) denotes the minimum number ofcops required to catch the robber in finite time. Westudy the cop number of geometric graphs. For pointsx1, x2, . . . , xn ∈ R2, and r ∈ R+, the geometric graphsG(x1, . . . , xn; r) is the graph on these n points, with xi, xj

adjacent when ‖xi − xj‖ ≤ r. We prove that c(G) ≤ 9for any connected geometric graph G ∈ R2. We improveon this bound for random geometric graphs that are suf-ficiently dense. Let G(n, r) denote the probability spaceof geometric graphs with n vertices chosen uniformly andindependently from [0, 1]2. For G ∈ G(n, r), we show thatwith high probability (whp), if nr4 � log n then c(G) ≤ 2and if nr4 � log n then c(G) = 1. Finally, we providea lower bound near the connectivity regime of G(n, r): ifnr2 � log2 n then c(G) > 1 whp.

Andrew J. BeveridgeDepartment of Mathematics, Statistics and ComputerScienceMacalester [email protected]

Andrzej DudekDepartment of MathematicsWestern Michigan [email protected]

Alan FriezeDepartment of Mathematical SciencesCarnegie Mellon [email protected]

Tobias MuellerCentrum voor Wiskunde en [email protected]

MS37

Seepage in Directed Acyclic Graphs

Seepage is a vertex-pursuit game played on directed acyclicgraphs (or dags), introduced by Nowakowski et al. Weconsider applications of Seepage to the modelling of hier-archical social networks. A stochastic model is introducedwhich generates dags with a prescribed degree sequence.We play a variant of Seepage on the model, and contrastresults on the green number in the case of a constant versusa power law degree sequence.

Anthony BonatoRyerson [email protected]

MS37

Ambush Cops and Robbers

A variation of the game with two robbers is introduced.The cops win by moving onto the same vertex as one of therobbers after a finite number of moves. The robbers winby avoiding capture indefinitely or by both moving ontothe same vertex as the cop. (Otherwise, the robbers are ondistinct vertices.) We present a recognition theorem (andcop strategy) for graphs on which one cop can guarantee awin. (Joint work with M. Creighton)

Nancy E. ClarkeAcadia [email protected]

MS37

Cops and Scared Robber

In a traditional game of Cops and Robber, the cops androbber move alternately on a graph, where each player canelect to move to an adjacent vertex or remain at his currentposition. In this case, the Scared Robber has the restrictionthat he can not move closer to any cop. The motivationfor adding this restriction is to gain insight into the rob-ber’s strategy in the traditional game, and determine whenmoving toward a cop would actually be to the robber’s ad-vantage.

Shannon L. FitzpatrickUniversity of Prince Edward [email protected]

Stephen FinbowMath, Stats, and CSSt. Francis Xaxier [email protected]

MS37

Chasing Cops on Random Graphs

In the game of cops and robber, the cops try to capture arobber moving on the vertices of the graph. The minimumnumber of cops required to win on a given graph G is calledthe cop number of G. The biggest open conjecture in thisarea is the one of Meyniel, which asserts that for someabsolute constant C, the cop number of every connected

graph G is at most C√|V (G)|. We show that Meyniel’s

conjecture holds asymptotically almost surely for a randomd-regular graph, d ≥ 3, as well as in the standard randomgraph modelG(n, p). (Joint work with Nick Wormald.) Wealso investigate the game played on a (percolated) random

DM12 Abstracts 85

geometric graph. (Joint work with Noga Alon.)

Pawel PralatRyerson [email protected]

Nick WormaldDepartment of Combinatorics and OptimizationUniversity of Waterloo, [email protected]

Noga AlonSchools of Mathematics and Computer ScienceTel Aviv [email protected]

MS38

An Analogue of the Harer-Zagier Formula for Gen-eral Surfaces

We consider the different ways of gluing the edges of a 2n-gon in pairs so as to create a surface without boundary. Upto homeomorphism, the surface obtained is characterizedby its genus and by its orientability. The Harer-Zagierformula characterizes the generating function of the genusof the orientable surfaces obtained from the gluings of the2n-gon. In this work we give an analogue of the Harer-Zagier for general (locally orientable) surfaces. The proofis bijective.

Olivier [email protected]

MS38

Rooted K2,4 Minors in Planar Graphs

Given 3 terminal vertices in a graph G, Robertson andSeymour characterized whether we could find a rooted K2,3

minor in G using the terminals for the larger side of thebipartition. We provide a characterization for when G has4 terminals and is planar.

Lino DemasiSimon Fraser [email protected]

MS38

Coloring Graphs on Surfaces

We will review new progress on coloring and list-coloringembedded graphs. This talk is based on works with Ken-ichi Kawarabayashi, Bernard Lidicky, Bojan Mohar, LukePostle and Robin Thomas.

Zdenek DvorakDepartment of Applied MathematicsCharles University, [email protected]

MS38

Clique Immersion in Digraphs

Immersion is a containment relation between graphs (ordigraphs) which is defined similarly to the more familiarnotion of minors, but is incomparable to it. This talkwill begin with a gentle introduction, and eventually showthat every Eulerian digraph with minimum degree at least

t(t − 1) contains a bidirected clique of order t as an im-mersion. Joint work with Matt DeVos, Bojan Mohar andDiego Scheide.

Jessica McDonaldSimon Fraser Universityjessica [email protected]

MS38

Hamiltonicity of Graphs on Surfaces

In 1956, Tutte proved that every 4-connected plane graphhas a hamiltonian cycle. Beginning with the result byTutte, many results on the hamiltonicity of graphs on sur-faces were shown. However, the following conjecture byGrunbaum and Nash-Williams has been unsolved for morethan 40 years; every 4-connected graph on the torus has ahamiltonian cycle. In this talk, we will mention recentresults around the conjecture by Grunbaum and Nash-Williams.

Ken-ichi Kawarabayashi, Kenta OzekiNational Institute of Informatics, Japank [email protected], [email protected]

MS39

Distinguishing Numbers and Regular Orbits

For a group G acting on a set X, the distinguishing numberis the least number of parts in a partition of X which ispreserved only by the identity element of G. Various resultsabout distinguishing numbers can be expressed in terms oforbits of G on the power set of X. We survey some knownresults from the group theory literature, and consider someproblems of a combinatorial nature.

Robert BaileyUniversity of [email protected]

MS39

On the Orders of Symmetric Graphs

In this talk I will describe some recent developments con-cerning finite symmetric (arc-transitive) graphs. The firstis the computer-assisted determination of all cubic (3-valent) symmetric graphs on up to 10000 vertices, whichproduced some new large graphs with given degree and di-ameter. The second is a new approach for classifying allconnected symmetric cubic graphs of given orders, show-ing for example that for any fixed k and any integer s > 1,there are only finitely many s-arc-transitive cubic graphsof order kp where p is prime. In turn, these results have ledto some further new discoveries about the orders of sym-metric graphs of higher valency (in some recent joint workwith Caiheng Li and Primoz Potocnik).

Marston ConderUniversity of Auckland, New [email protected]

MS39

A Catalog of Self-Dual Plane Graphs with Max De-gree 4

In this paper, we produce a catalog of self-dual planegraphs with maximum degree 4 (self-dual spherical grids).Based on their automorphism groups, self-dual spherical

86 DM12 Abstracts

grids fall into a finite number of parameterized, infinitefamilies. The individual self-dual spherical grids in a fam-ily have the same basic shape, differing only in size.

Elizabeth HartungMassachusetts College of Liberal [email protected]

Jack GraverMathematics DepartmentSyracuse [email protected]

MS39

Hamiltonian Cycles in Cayley Graphs

I will provide an overview of general results on hamilto-nian cycles in Cayley graphs, focusing on some recent workthat deals with graphs whose order has few prime factors.Some of this is based on work produced jointly in variouscombinations with Steve Curran, Klavdija Kutnar, DraganMarusic, Dave Witte Morris, and Primoz Sparl.

Joy MorrisMathematics and Computer ScienceUniversity of [email protected]

MS40


Abstract not avaialable at time of publication.

Joel S. BaderJohns Hopkins UniversityDepartment of Biomedical [email protected]

MS40

Network Archaeology: Uncovering Ancient Net-works from Present-Day Interactions


Carl KingsfordDepartment of Computer ScienceUniversity of Maryland, College [email protected]

MS40

Understanding Phenome-genome Association byCo-clustering and Graph Matching

Graph algorithms are playing an increasingly importantrole in studying phenotype-gene relations in the contextof systems biology and phenome (the whole collection ofphenotypes) analysis. I will introduce two methods thatutilize network information in phenome network and gene-relation network to study phenome-genome associations, arandom-walk based graph-matching algorithm (Bi-RW) forunveiling associations between phenome and genes, and aregularized NMF algorithm (R-NMTF) to co-cluster phe-notypes and genes, and detect associations between phe-notype clusters and gene clusters.

Rui KuangDept Computer ScienceUniversity of Minnesota

[email protected]

MS40

Sensitive Graph-theoretic Approaches for Extract-ing Function from Biological Networks

Sequence-based computational approaches have revolu-tionized our understanding of biology. Since proteins ag-gregate to perform function instead of acting in isolation,the connectivity of a protein interaction network (PIN) willdeepen biological insights over and above sequences of in-dividual proteins. The advancement relies on developingsensitive graph-theoretic methods for extracting biologicalknowledge from PINs. Analogous to sequence alignment,alignment of PINs will impact biological understanding.Network alignment maps nodes of different networks withthe goal of ”fitting” one network into the other well toidentify topologically similar network regions and transferfunction between them. Since network alignment is com-putationally infeasible, owing to the NP-completeness ofthe underlying subgraph isomorphism problem, heuristicalgorithms are sought. We designed a sensitive measure ofthe topological position of a node in the network, graphletdegree vector (GDV), to measure the number of 2-5-nodegraphlets (induced subgraphs) that the node participatesin. Hence, GDV of a node is a detailed topological de-scriptor of its extended network neighborhood. Our GDV-similarity measure compares GDVs of two nodes to quan-tify similarity of their extended networks neighborhoods.Using these measures, we developed two network alignmentalgorithms: a greedy ”seed and extend” approach thatquickly finds approximate alignments and an approach thatfinds optimal alignments by using the ”expensive” Hungar-ian algorithm. Their alignments expose surprisingly largeregions of network and functional similarity even in dis-tant species, suggesting broad similarities in cellular wiringacross all life on Earth. Also, we used these measures toshow that aging, cancer, pathogen-interacting, drug-targetand genes involved in signaling pathways are ”central” inthe network, occupying dense network regions and ”domi-nating” other genes in the network.

Tijana MilenkovicDepartment of Computer Science and EngineeringUniversity of Notre [email protected]

MS40

Biological Networks in Three Dimensions

Two studies are highlighted where three-dimensional mod-eling of protein networks provides insights into the bio-physics and evolution of biological systems. First, a struc-tural analysis of the yeast protein-protein interaction net-work reveals that protein evolution is constrained by thebiophysics of protein folding, the mutational robustness ofproteins, and the biophysics and function of protein-proteininteractions. Second, a structural analysis of human-virusprotein-protein interaction networks reveals distinct prin-ciples governing antagonism versus cooperation in host-pathogen interactions.

Yu (Brandon) Xia

Bioinformatics Program & Dept. of ChemistryBoston [email protected]

DM12 Abstracts 87

MS41

Advances on Quantifier Elimination and Applica-tions

In this talk, we present a new approach for computingcylindrical algebraic decomposition of real space via trian-gular decomposition of polynomial systems. Based on it,a general quantifier elimination method is proposed. Theusage of our method is illustrated by several applicationexamples.

Changbo ChenUniversity of Western [email protected]

Marc Moreno MazaComputer Science DepartmentUniversity of Western [email protected]

MS41

Automatic Quantifier Elimination Proves a KeyResult in Submodular Function Minimization

We show how symbolic computation can automaticallyprove an important and well-known result in combinatorialoptimization. Specifically, we address the question “Whichobjective functions can be minimized by reduction to s-tmin cut?” This question has been answered by a number ofhighly-cited papers in discrete mathematics and computervision. By posing this same question to a computer algebrasystem, we automatically identify submodularity as beingthe key condition for reduction to s-t min cut.

Andrew DelongThe University of Western Ontario, [email protected]

MS41

On Newton Polytopes, Tropisms, and Puiseux Se-ries to Solve Polynomial systems

Sparse polynomial systems in several variables are charac-terized by a set of exponents of monomials that appear withnonzero coefficient. The convex hull of this set of exponentsis the Newton polytope. To a system of sparse polynomialequations corresponds a tuple of Newton polytopes. Vec-tors perpendicular to a tuple of edges of all polytopes aretropisms when they define the leading powers of a Puiseuxseries development for a solution of the polynomial sys-tem. In this talk we outline a polyhedral approach to solvepolynomial systems by means of Puiseux series. On stan-dard benchmark problems as the cyclic n-roots systems weobtained exact representations for surfaces of solutions.

Jan Verschelde, Danko AdrovicDepartment of Mathematics, Statistics and ComputerScienceUniversity of Illinois at [email protected], [email protected]

MS41

Algebraic Representations of Branches of Func-tions

Apparently-simple function definitions may hide subtletiesfrom branch cuts in the complex plane. Hence formulaethat initially seem correct may fail for various values. One

remedy is to consider the complement of the branch cuts,decomposing this space by Cylindrical Algebraic Decompo-sition and checking validity on each connected component.This process will be discussed and how it is possible to im-prove this algorithm’s efficiency with pre-conditioning andchoice of variable order.

David J. Wilson, Russell Bradford, James DavenportUniversity of [email protected], [email protected],[email protected]

MS41

A Solver for Linear Algebraic Systems Resultingfrom Selecting Discrete Structures

A computer algebra program LSSS (Linear Selective Sys-tems Solver) has been developed to solve linear systemsefficiently that are sparse, overdetermined and have manyvariables that take the value zero in the solution. Ap-plications include the determination of first integrals andLie-symmeties of systems of non-commutative ODEs re-quiring the solution of linear systems with over one billionequations, the computation of Poisson homologies and co-homologies of Poisson algebras and the study of compatiblePoisson structures.

Thomas WolfDepartment of MathematicsBrock [email protected]

MS42

A De Bruijn - Erdos Theorem in ConnectedGraphs?

De Bruijn and Erdos proved that every noncollinear set ofn points in the plane determines at least n distinct lines.Chen and Chvatal suggested that this might generalize toall finite metric spaces with appropriately defined lines.As for metric spaces defined by connected graphs, the ana-logue of the De Bruijn - Erdos theorem is known to holdfor all bipartite graphs, all chordal graphs, and all graphsof diameter two.

Vasek ChvatalConcordia [email protected]

MS42

Matthews Sumner Conjecture on Claw-FreeGraphs

The Matthews Sumner Conjecture is that every 4-connected claw-free graph is Hamiltonian. There are alarge number of conjectures, some seemingly weaker andsome seemingly stronger, that are equivalent to this con-jecture that will be discussed. For example, the CarstenThomassen Conjecture that every 4-connected line graphis Hamiltonian is an equivalent conjecture.

Ralph FaudreeUniversity of MemphisDepartment of Math [email protected]

88 DM12 Abstracts

MS42

How to Recognize a Good Conjecture

The speaker will present some challenges in dealing withopen problems and conjectures. He may discuss some spe-cific graph theory conjectures and demonstrate some ratio-nale behind the Open Problem Garden.

Bojan MoharSimon Fraser [email protected]

MS42

Some of my favorite Conjectures

We will present and briefly discuss some open questions intwo areas: (1) Cyclic Coloration of plane graphs and (2)Well-covered graphs.

Michael D. PlummerDepartment of MathematicsVanderbilt [email protected]

MS42

The Implicit Representation of Graphs Conjecture

A class of graphs P is called a hereditary property ifit is closed under taking induced subgraphs and isomor-phism. Common examples include being acyclic, beingplanar, and being bipartite. Given a hereditary propertyP , we seek a method of labeling the vertices of graphs inP with short labels [that is, O(log n) bits for each ver-tex] such that we can determine adjacency of two ver-tices just by considering their labels. Formally, a prop-erty P admits an implicit representation provided thereis a function A : Z+ × Z+ → {0, 1} and a positive inte-ger k such that for every G ∈ P with there is a mapping� : V (G) → [nk] (where n = |V (G)|) such that u ∼ v ifand only if A(�(u), �(v)) = 1. It is easy to create sucha labeling for acyclic graphs: Pick a root vertex in eachcomponent, number the vertices from 1 to n, and labelvertex v as (v, w) where w is the unique neighbor of v ona path from v to the root (or use the label (v, v) if v isthe root). However, bipartite graphs admit no such repre-sentation simply because there are too many. That is, fora property P , the speed of P is a function that gives foreach n the number of labeled graphs in P with V = [n].If a property has an implicit representation, then neces-sarily its speed is bounded by an expression of the formncn for a positive constant c. The Implicit RepresentationConjecture of Kannan, Naor, and Rudich posits that thisnecessary condition is also sufficient.

Ed ScheinermanJohns Hopkins [email protected]

MS43

Ramsey Classes Defined by Forbidden Homomor-phisms

I will talk about Ramsey properties of classes of digraphs(or more generally, relational structures) defined by forbid-ding homomorphisms from a fixed set of digraphs. Caseswith infinitely many forbidden structures are of particu-lar interest. These classes have several interesting applica-

tions.

Jan FoniokQueen’s [email protected]

MS43

A Randomized Version of Ramsey’s Theorem

For every integer r ≥ 2 we call a k-uniform hypergraphH on n vertices r-Ramsey-forcing if every r-edge-coloringof Kn contains a monochromatic copy of Kk whose ver-tices form a hyperedge in H . We determine the thresholdfor a random k-uniform hypergraph on n vertices and edgeprobability q to be r-Ramsey-forcing. Further we discusscommon features of the above problem with the more clas-sical Ramsey’s theorem for random (hyper-)graph G(n, p).

Luca GugelmannInstitute of Theoretical Computer Science ETH Zurich8092 Zurich, [email protected]

Yury PersonFreie Universitat, [email protected]

Angelika StegerInstitute of Theoretical Computer ScienceETH, [email protected]

Henning ThomasInstitute of Theoretical Computer Science ETH Zurich8092 Zurich, [email protected]

MS43

General Deletion Lemmas via the Harris Inequality

For a given graph H let XH denote the random variablethat counts the number of copies of H in a random graphGn,p. In order to control the upper tail of XH , Rodl andRucinski showed that with ‘Janson-like’ probability, delet-ing a small fraction of all edges suffices to reduce the num-ber of copies of H to at most (1+ε)E[XH ]. This approachis known as the ‘deletion method’ and appears in manyproofs of Ramsey properties of random structures. In thistalk I will present a variant of the deletion method that isbased on the Harris inequality and extends to more generalgraph properties.

Reto SpohelMax-Planck-Institut fur [email protected]

Angelika StegerInstitute of Theoretical Computer ScienceETH, [email protected]

Lutz WarnkeUniversity of [email protected]

DM12 Abstracts 89

MS43

On Ramsey Multiplicities of Graphs ContainingTriangles

A graph G is k-common (k > 1) if the minimum number ofmonochromatic copies of G over all k edge colorings of Kn

is asymptopic to the expected number of monochromatic

copies of G in a random k edge coloring. Jagger, Stovıcekand Thomason defined the class of k-common graphs, andshowed among other results that every graph containingK4 as a subgraph is not 2-common. We prove that everygraph containing K3 as a subgraph is not 3-common.

Michael YoungIowa State [email protected]

James CummingsCarnegie Mellon [email protected]

MS44

Multidimensional Algorithmic Mechanism Design

In his seminal paper, Myerson [1981] provides a revenue-optimal auction for a seller who is looking to sell a singleitem to multiple bidders. Extending this auction to simul-taneously selling multiple heterogeneous items has beenone of the central problems in Mathematical Economics.We provide such an extension that is also computationallyefficient.

Costis DaskalakisMassachusetts Institute of [email protected]

MS44

Combinatorial Walrasian Equilibria

One way to resolve a combinatorial auction is via Wal-rasian equilibrium: setting a price for each item so that,when bidders take their most demanded sets, each objectis chosen by exactly one bidder. Such equilibria have de-sirable properties, but they exist only in special cases. Inthis work we introduce the alternative notion of combina-torial Walrasian equilibrium, in which one first partitionsthe objects into indivisible bundles and then sets a price foreach bundle. We design mechanisms that generate combi-natorial Walrasian equilibria for various classes of biddervaluations, with the goal of maximizing social welfare. Ourstudy also touches upon revenue and truthfulness in suchmechanisms.

Brendan LucierMicrosoft Research New [email protected]

Michal FeldmanHebrew University of [email protected]

MS44

Knightian Auctions

In an auction, a player may not exactly know his own val-uation, or its distribution. We thus study single-good auc-tions whose players know their own valuations only withina multiplicative factor (e.g., 10%). The notions of im-

plementation in dominant and undominated strategies arenaturally extended to this setting, but their power is vastlydifferent. Namely, (1) We prove that no dominant-strategymechanism can guarantee more social welfare than by as-signing the good at random; but (2) We prove a much bet-ter performance can be obtained via undominated-strategymechanisms, and actually provide tight upper and lowerbounds for the fraction of the maximum social welfareguaranteable by such mechanisms, whether deterministicor probabilistic.

Silvio MicaliMassachusetts Institute of [email protected]

Alessandro Chiesa, Zeyuan [email protected], [email protected]

MS44

Online Procurement

We present results for online procurement markets. Weintroduce mechanisms with desirable approximation guar-antees in various models of online procurement. We alsopresent new models of strategic bidding in such markets,and discuss new mechanism design frameworks for thesebidding models.

Yaron SingerUniversity of California at [email protected]

MS44

Bayesian Multi-Parameter Scheduling

We study the makespan minimization problem with unre-lated selfish machines. Specifically, our goal is to schedulea given set of jobs on m machines; the machines are strate-gic agents who hold the private information of the run-timeof each job on them. The goal is to design a mechanismthat minimizes makespan — the time taken for the last jobto complete. In the strategic setting, strong impossibilityresults are known that show that no truthful (anonymous)mechanism can achieve better than a factor m approxima-tion. We show that under mild Bayesian assumptions it ispossible to circumvent such negative results and obtain aconstant approximation.

Balasubramanian SivanUniversity of [email protected]

Shuchi ChawlaDepartment of Computer SciencesUniversity of [email protected]

Jason HartlineNorthwestern [email protected]

David MalecUniversity of [email protected]

90 DM12 Abstracts

MS45

Graph Homomorphism Counts Mod M

I will talk about the numbers hom(F,G) mod m as thegraphs F and G vary.

Swastik KoppartyMassachusetts Institute of [email protected]

MS45

Frozen Vertices in Colourings of a Random Graph

Over the past decade, much of the work on random k-SAT,random graph colouring, and other random constraint sat-isfaction problems has focussed on some foundational un-proven hypotheses that have arisen from statistical physics.Some of the most important such hypotheses concern the“clustering’ of the solutions. It is believed that if the prob-lem density is sufficiently high then the solutions can bepartitioned into clusters that are, in some sense, both well-connected and well-separated. Furthermore, the clusterscontain a linear number of “frozen variables”, whose val-ues are fixed within a cluster. The density where suchclusters arise corresponds to an algorithmic barrier, abovewhich no algorithms have been proven to solve these prob-lems. In this talk, we prove that frozen vertices do indeedarise for k-colourings of a random graph, when k is a suffi-ciently large constant, and we determine the exact densitythreshold at which they appear.

Michael MolloyUniversity of [email protected]

MS45

Colorings and Homomorphisms of Sparse Graphs

chromatic number and low tree-width / low tree-depth col-orings. Dichotomy of sparsity by means of chromatic num-bers. Applications to distance coloring, circular chromaticnumber of large girth graphs with sub-exponential expan-sion, etc.

Jaroslav Nesetrildepartment of Applied MathematicsCharles [email protected]

Patrice Ossona de MendezCNRS ParisCentre d’Analyse et de Mathematiques [email protected]

MS45

Fast Algorithms for Sparse Graphs

Algorithms for sparse graphs: subgraph isomorphism prob-lem, connection with bounded local tree-width. Low TreeDepth Decomposition and first-order properties. Applica-tion to graph invariants.

Patrice Ossona de MendezCNRS ParisCentre d’Analyse et de Mathematiques [email protected]

Jaroslav Nesetril

department of Applied MathematicsCharles [email protected]

MS45

Semilattice and NU Polymorphisms on ReflexiveGraphs

The structures H for which arc-consistency solves H-COLare exactly those that admit TSI polymorphisms. The twomain sources of TSI polymorphisms are near unanimity(NU) and semilattice (SL) polymorphisms. Much has beendone to understand the nature of NU polymorphisms, butSL polymorphisms are not so well understood. In this talkwe describe the structure of symmetric reflexive graphs ad-mitting various types of SL polymorphisms. Further we re-late the existence of SL polymorphisms on these graphs tothe existence of NU polymorphisms of various types, andto related classes of graphs. This is preliminary work in inunderstanding SL polymorphisms, and includes joint workwith Pavol Hell.

Mark SiggersKyungpook National UniversityDaegu, Republic of [email protected]

MS46

Near Factorisations of the Complete Graph

A w-near k-factor of a graph G on n vertices is a spanningsubgraph of G with w vertices of degree 0 and n− w ver-tices of degree k. In this talk, we introduce the concept ofa w-near k-factorization of a graph G, which is a decom-position of G into w-near k-factors. Thus, for example,a k-factorization is equivalent to a 0-near k-factorization,and a near 1-factorization is equivalent to a 1-near 1-factorization. We focus on w-near 2-factorizations of thecomplete graph Kn and Kn−I ; in the case that the partial2-factors are required to be pairwise isomorphic, this maybe viewed as a generalization of the Oberwolfach problem.We discuss some constructions of w-near 2-factorizations inwhich all cycles in the near-factors have the same length.

Peter Danziger, Andrea BurgessRyerson [email protected], [email protected]

MS46

Bounds for Covering Arrays with Row Limit

Covering arrays with row limit (CARLs) are a generaliza-tion of covering arrays. They have a new parameter rowweight, w, representing the number of non-empty cellsin a row. Considering w as a function of the number ofcolumns, k, there are two extremal cases: when w is aconstant, CARLs are equivalent to group divisible cover-ing designs and we know their optimal size; when w = k,CARLs are covering arrays, and their optimal size is stillunknown in most cases. We present two probabilistic mod-els to determine the upper bounds on the size of a CARLwhen w = f(k) and we look into critical points where thechange in the behaviour occurs.

Nevena FranceticUniversity of [email protected]

DM12 Abstracts 91

Peter DanzigerRyerson [email protected]

Eric MendelsohnUniversity of TorontoDepartment of [email protected]

MS46

A new construction of strength-3 covering arraysusing primitive polynomials over finite fields

In this talk, we present a new construction of covering ar-rays based on Linear Feedback Shift Register (LFSR) se-quences constructed using primitive polynomials over finitefields. For any prime power q, this construction gives a cov-ering array of strength 3 with q2 + q + 1 factors/columnsover q levels/symbols that has size 2q3−1 (number of rows).The construction can be extended to non-prime powers vby dropping levels from a larger prime power. This re-sults in significant reductions on known upper bounds forcovering array sizes in most cases covered by this construc-tion. In particular, for the values of v ≤ 25 kept in Col-bourn’s covering array tables, this construction improvesupper bounds for all v �= 2, 3, 6.

Lucia MouraUniversity of OttawaInformation [email protected]

Sebastian RaaphorstUniversity of [email protected]

Brett C. StevensCarleton UniversitySchool of Mathematics and [email protected]

MS46

New Areas in Covering Arrays

In recent years, with Gary Bazdell, I have had increasedcommunication and collaboration with industrial usersof combinatorial designs, predominantly covering arrays.Coming out of their needs and interests, Gary and I havebeen motivated to think about new generalisations of cov-ering arrays, specifically ones that introduce order to therows and columns. I will discuss these definitions, themathematics and their connections to applications.

Brett C. StevensCarleton UniversitySchool of Mathematics and [email protected]

MS46

Nonincident Points and Blocks in Designs

We study the problem of finding the largest possible setof s points and s blocks in a balanced incomplete blockdesign, such that that none of the s points lie on any ofthe s blocks. We investigate this problem for two typesof BIBDs: projective planes and Steiner triple systems.For a projective plane of order q, we prove that s ≤ 1 +(q + 1)(

√q − 1) and we also show that equality can be

attained in this bound whenever q is an even power oftwo. For a Steiner triple system on v points, we prove thats ≤ (2v + 5 − √24v + 25)/2 and we determine necessaryand sufficient conditions for equality to be attained in thisbound.

Douglas R. StinsonUniversity of WaterlooDavid R. Cheriton School of Computer [email protected]

MS47

Optimality of the Neighbor Joining Algorithm andFaces of the Balanced Minimum Evolution Poly-tope

Balanced minimum evolution (BME) is a statistically con-sistent distance-based method to reconstruct a phyloge-netic tree from an alignment of molecular data. In 2000,Pauplin showed that the BME method is equivalent to op-timizing a linear functional over the BME polytope, theconvex hull of the BME vectors obtained from Pauplinsformula applied to all binary trees. The BME method isrelated to the popular Neighbor Joining (NJ) algorithm,now known to be a greedy optimization of the BME prin-ciple. In this talk I will elucidate some of the structureof the BME polytope and strengthen the connection be-tween the BME method and NJ Algorithm. I will showthat any subtree-prune-regraft move from a binary tree toanother binary tree corresponds to an edge of the BMEpolytope. Moreover, I will describe an entire family offaces parametrized by disjoint clades. Finally, given a phy-logenetic tree T, I will show that the BME cone and everyNJ cone of T have intersection of positive measure.

David HawsUniversity of [email protected]

Terrell HodgeWestern Michigan [email protected]

Ruriko YoshidaUniversity of [email protected]

MS47

Finding the Max Cut of the Genetic InteractionsGraph in Yeast Finds Compensatory Pathways.

By using a randomized local search method to find the maxcut in the graph of genetic interactions in yeast we generatepotential compensatory networks. Unlike other methods,this method works solely on genetic interaction data fromepistatic miniarray profiles (E-MAP) and synthetic geneticarray analysis (SGA), this enables us to use physical inter-actions as an additional validation source. We finish withsome open questions regarding the behavior of our tech-nique on different graphs and weights.

Benjamin HescottTufts [email protected]

Mark LeisersonBrown [email protected]

92 DM12 Abstracts

Andrew Gallant, Diana TatarTufts [email protected], [email protected]

Lenore CowenDept. of Computer ScienceTufts [email protected]

MS47

Statistics and Applications of Tree Space

The space of metric n-trees is a polyhedral complex, inwhich each polyhedron corresponds to a different treetopology. Under the construction of Billera, Holmes, andVogtmann, this space is non-positively curved, so there isa unique geodesic (shortest path) between any two trees.We will look at what it means to do statistics on this space,as well as some biological applications.

Megan OwenFields [email protected]

MS47

Comparing Biological and Mathematical Ap-proaches to Modeling Tissue Development in C.elegans

Gene networks are often inferred from experimental datausing single-method approaches; however, an individualmethod may not sufficiently recover the network. We de-veloped a modeling pipeline that combines statistical andalgebraic methods. We built a mathematical model of tis-sue development from wildtype C. elegans data. We com-pared it to an existing biological model on a gold-standardnetwork. While both models identified many interactions,the mathematical model was enriched for positive predic-tions compared to biological model.

Brandilyn StiglerSouthern Methodist [email protected]

MS47

Profiling RNA Secondary Structures

Efficient methods for sampling from the Boltzmann distri-bution have led to an emergence of ensemble-based ap-proaches to RNA secondary structure prediction. Wepresent a novel combinatorial method, RNA structure pro-filing, for identifying patterns in structural elements acrossa Boltzmann sample. Our approach is based on classifyingstructures according to features chosen from well-definedstructural units called helix classes. We show that combi-natorial profiling is straightforward, stable and surprisinglycomprehensive.

Emily Rogers, M. Shel SwensonSchool of MathematicsGeorgia Institute of [email protected], [email protected]


MS48

New Results on D-Optimal Matrices

D-optimal matrices are 2v 2v (-1,+1)-matrices that havemaximal determinant among all 2v 2v (-1,+1)-matrices,where v is an odd positive integer. The value of the max-imal determinant is given by Ehlich’s bound. We presentnew theoretical and computational results on D-optimalmatrices of circulant type. Such D-optimal matrices areconstructed via two circulant submatrices of orders v each.In particular, we construct new D-optimal matrices of or-ders 206, 242, 262, 482. Joint work with D. Z. Djokovic.

Ilias S. KotsireasWilfrid Laurier UniversityDepartment of [email protected]

MS48

Matroid Base Polytope Decomposition and Appli-cations

Let P (M) be the matroid base polytope of a matroid M . Amatroid base polytope decomposition of P (M) is a decom-

position of the form P (M) =t⋃

i=1

P (Mi) where each P (Mi)

is also a matroid base polytope for some matroid Mi, andfor each 1 ≤ i �= j ≤ t, the intersection P (Mi) ∩ P (Mj)is a face of both P (Mi) and P (Mj). In this talk, we shall

present some new results on the above decomposition. Wegive sufficient conditions on M in order that P (M) has ahyperplane split (that is, a decomposition with t = 2) andshow that P (M) has not a hyperplane split if M is binary.We finally discuss an application in relation with Tutte andEhrhart polynomials.

Jorge L. Ramirez AlfonsinMontpellier 2, [email protected]

MS48

Generating Program Invariants via Interpolation

This talk focuses on automatically generating polynomialequations that are inductive loop invariants of computerprograms. A new method which is based on polynomialinterpolation is proposed. Though the proposed method isnot complete, it is efficient and can be applied to a broaderrange of problems compared to existing methods targetingsimilar problems. The efficiency of our approach is testifiedby experiments on a large collection of programs.

Rong XiaoUniversity of Western [email protected]


MS48

Parallel Computation of the Minimal Elements ofa Poset and Applications

Computing the minimal elements of a partially ordered fi-nite set (poset) is a fundamental problem in combinatorics

DM12 Abstracts 93

with numerous applications. In a previous work, we havedeveloped for this task a divide-and-conquer algorithmwhich is also cache-efficient and which can be efficientlyparallelized free of determinacy races. We adapted thistechnique to questions arising in computer algebra (polyno-mial expression optimization) and combinatorics (transver-sal hypergraph generation). In this talk, we present newapplications in both computer algebra (coprime factoriza-tion) and combinatorics (orthogonal hypergraph genera-tion). We have implemented the corresponding algorithmsin Cilk++ targeting multi-cores. For our test problems ofsufficiently large input size our code demonstrates a linearspeedup on 32 cores.

Yuzhen XieUniversity of Western [email protected]


MS49

Conjectures on Thickness, Connectivity, and OtherThings

Various conjectures that I’ve found intriguing will be dis-cussed. These will include a conjecture on thickness andchromatic number, a mixed-connectivity conjecture, andthe total graph conjecture.

Lowell BeinekeIndiana University - Purdue University Fort [email protected]

MS49

Snarks and Their Relation to Conjectures aboutGraphs

In this talk I will shortly discuss the role of snarks in prov-ing or refuting conjectures and sketch an algorithm thatwas used to generate large lists of snarks. The results ofthese computations and some of the published conjecturesthat could be refuted by the snarks generated will be pre-sented. This research is joint work with Jan Goedgebeur,Jonas Hagglund and Klas Markstrom

Gunnar BrinkmannUniversity of Ghent,[email protected]

MS49

On Conjectures of Grafitti.pc

This talk will begin with a brief description of how theconjecture-making computer program, Graffiti.pc, gener-ates graph theoretical conjectures. Then, I will talk aboutsome of my favorite Graffiti.pc conjectures.

Ermelinda DeLaVinaUniversity of [email protected]

MS49



Fan Chung GrahamUniversity of California, San [email protected]

MS51

On Metric Dimension of Functigraphs

A set S of vertices in a graph G is called a resolving set forG if for every pair of vertices x, y ∈ V (G), there is a vertexs ∈ S such that d(x, s) �= d(y, s). The minimum numberof vertices in a resolving set for G is the metric dimensiondim(G) of G. Given two copies G1 and G2 of a graph G anda function f : V (G1)→ V (G2), the functigraph C(G, f) isthe graph with vertices V (G1)∪V (G2) and edges E(G1)∪E(G2) ∪ {xy|y = f(x)}. In this talk, we give bounds onthe metric dimension of functigraphs, particularly in thecases when G is a cycle or path and when the function fis a constant function or permutation.

Linda ErohDepartment of MathematicsUniversity of Wisconsin at [email protected]

Cong Kang, Eunjeong YiTexas A&M University at [email protected], [email protected]

MS51

Closed K-Stop Distance in Graphs.

The Traveling Salesman Problem (TSP) is still one of themost researched topics in computational mathematics, andwe introduce a variant of it, namely the study of the closedk-walks in graphs. We search for a shortest closed routevisiting k cities in a non complete graph without weights.This motivates the following definition. Given a set of kdistinct vertices S = {v1, v2, . . . , vk} in a simple graph G,the k-circuit-distance of set S is defined to be

dk(S) = minθ∈P(S)

(d(θ(x1), θ(x2))+d(θ(x2), θ(x3))+. . .+d(θ(xk), θ(x1))

),

where P(S) is the set of all permutations from S onto S .That is the same as saying that dk(S) is the length of theshortest circuit through the vertices {x1, . . . , xk}. Recallthat the Steiner distance sd(S) is the number of edges in aminimum connected subgraph containing all of the verticesof S . We note some relationships between Steiner distanceand circuit distance. The 2-circuit distance is twice theordinary distance between two vertices. We conjecture thatradk(G) ≤ diamk(G) ≤ k

k−1radk(G) for any connected

graph G for k ≥ 2. For k = 2, this formula reduces to thewell-known classical result rad(G) ≤ diam(G) ≤ 2rad(G).We prove the conjecture in the cases when k = 3 and k = 4for any graph G and for k ≥ 3 when G is a tree.

Ralucca M. GeraNaval Postgraduate [email protected]

Grady Bullington, Linda Eroh, Steve WintersUniversity of Wisconsin Oshkoshbullington@uwosh, eroh@uwosh, winters@uwosh

94 DM12 Abstracts

MS51

Distance in Rainbow Connected Graphs

A path (not necessarily a shortest one) between two ver-tices in a connected graph is rainbow-colored if each edgeof the path is labeled with a distinct color. A connectedgraph is rainbow-connected if the edges of the graph areassigned colors so that a rainbow-colored path exists be-tween every pair of vertices. In this paper, I discuss therecent work on methods for determining rainbow-coloredpaths in a given rainbow-connected graph.

Garry L. JohnsSaginaw Valley State [email protected]

MS51

Using Graphs for Facility Location Problems

The eccentricity of a vertex v in a connected graph G isthe distance from v to a vertex farthest from v in G. Thedistance of a vertex v in a connected graph G is the sumof the distances from v to the vertices of G. In this talk,we will consider subgraphs induced by these concepts forfacility location problems.

Steven J. WintersUniversity of Wisconsin [email protected]

MS52

Near-Unanimity Graphs and Absolute Retracts

We describe a generating set for the variety of reflexivegraphs that admit a k-ary near-unanimity operation; wefurther delineate a very simple subset that generates thevariety of absolute retracts with respect to j-holes; in par-ticular we show that every reflexive graph with a 4-NUoperation is an absolute retract for 3-holes. Our resultsgeneralise and encompass several results on NU-graphs andabsolute retracts from the past 30 years.

Tomas FederStanford [email protected]

Pavol HellSchool of Computer ScienceSimon Fraser [email protected]

Benoit LaroseConcordia [email protected]

Cynthia LotenFraser [email protected]

Mark SiggersKyungpook National UniversityDaegu, Republic of [email protected]

Claude TardifDepartment of Mathematics and Computer ScienceRoyal Military [email protected]

MS52

Graph Partitions

We consider partitions of the vertices of a graph into afixed number of labelled cells such that (i) the subgraphinduced by each cell belongs to a family that depends onthe cell, and (ii) edges joining vertices in different cells areallowed only if the cells correspond to adjacent vertices ina given pattern graph H . Both polynomiality and NP-completeness results are presented.

Gary MacgillivrayMathematics and StatisticsUniversity of [email protected]

Peter DukesUniversity of Victoria, [email protected]

Steve LowdonMathematics and StatisticsUniversity of [email protected]

MS52

Graph Partitions With Emphasis on 2K2-Partition

Pavol Hell will present some recent progress and someopen questions on the problem of partitioning a graph,or digraph, according to a prescribed pattern of adjacen-cies. Both classifications by complexity and characteriza-tions by forbidden induced subgraphs will be discussed.Then Daniel Paulusma will describe a recent solution (jointwith Barnaby Martin) of the problem of complexity ofthe disconnected cut, or equivalently the problem of 2K2-partition. Specifically, for a connected graph G = (V,E),a subset U of V is called a disconnected cut if both Uand V disconnect G. The problem to decide whether agraph has a disconnected cut is shown to be NP-complete.This problem is polynomially equivalent to the followingproblems: testing if a graph has a 2K2-partition, testingif a graph allows a vertex-surjective homomorphism to thereflexive 4-cycle and testing if a graph has a spanning sub-graph that consists of at most two bicliques. Hence, as animmediate consequence, these three decision problems areNP-complete as well.

Pavol HellSchool of Computer ScienceSimon Fraser [email protected]

Barnaby Martin, Daniel PaulusmaDurham [email protected],[email protected]

MS52

Bipartite Graphs and Approximation of MinimumCost Homomorphisms

For a fixed target graph H , the minimum cost homomor-phism problem, MinHOM(H), asks, for a given graph Gwith integer costs ci(u), u ∈ V (G), i ∈ V (H), and an in-teger k, whether or not there exists a homomorphism of Gto H of cost not exceeding k. When the target graph His a bipartite graph, a dichotomy classification is known:MinHOM(H) is solvable in polynomial time if and only

DM12 Abstracts 95

if H is a proper interval bigraph (i.e., a bipartite per-mutation graph). We suggest an integer linear program(ILP) formulation for MinHOM(H), and by rounding theILP solution, we obtain a 2-approximation algorithm forMinHOM(H), when H is a doubly convex bipartite graph.If time permits, we explain how to modify the roundingprocedure to obtain a |V (H)| - approximation algorithmfor MinHOM(H) when the complement of the bipartitegraph H is a circular arc graph.

Arash RafieySimon Fraser [email protected]

MS52

Adjoint Functors in Graph Theory

Functors in graph theory are essentially constructions thatmake new graphs out of input graphs. Two functors L, Rare respectively right and left adjoint to the other if there isa correspondence between the homomorphisms of L(G) toH and the homomorphisms of G to R(H). There are var-ious places where adjoint functors arise naturally withoutnecessarily being noticed: - Viewing lexicographic prod-ucts by complete graphs as right adjoints naturally definesKneser graphs as (cores of) left adjoints of complete graphs.- Reformulating the question of Harvey and Murty on theexistence of k-chromatic graphs with strongly independentcolour classes in terms of adjoint functors naturally leadsto the approach taken by Gyarfas, Jensen and Stiebitz toanswer it affirmatively. In the case of directed graphs andconstraint satisfaction problems, the adjoint functors arisein the arc graph construction, and they preserve tree dual-ity and many Maltsev-type operations related to complex-ity classifications. All of this suggests that the power andlimitations of adjoint functors in graph theory should bebetter understood.

Claude TardifDepartment of Mathematics and Computer ScienceRoyal Military [email protected]

Jan FoniokQueen’s [email protected]

MS53

New Results and Open Problems on Random Ge-ometric Graphs


Josep DiazDepartament de Llenguatges i Sistemes InformaticsUniversitat Politecnica de Catalunya, [email protected]

MS53

Modelling Interference in Wireless Networks usingGeometric Graphs

Given a set of positions for wireless nodes, the interferenceminimization problem is to assign a transmission radius(equivalently, a power level) to each node such that the re-sulting communication graph is connected, while minimiz-ing the maximum interference. In this talk I will discussrecent geometric graph models for representing interfer-

ence in ad hoc and sensor wireless networks, along withcorresponding algorithmic and complexity results for theinterference minimization problem.

Stephane DurocherUniversity of [email protected]

MS53

On the Treewidth of Random Geometric Graphs

We give asymptotically exact values for the treewidthtw(G) of a random geometric graph in [0,

√n]2. More pre-

cisely, we show that there exists some c1 > 0, such that forany constant 0 < r < c1, tw(G) = Θ( log n

log log n), and also,

there exists some c2 > c1, such that for any r = r(n) ≥ c2,tw(G) = Θ(r

√n). Our proofs show that for the corre-

sponding values of r the same asymptotic bounds also holdfor the pathwidth and treedepth of a random geometricgraph.

Dieter MitscheRyerson [email protected]

Guillem PerarnauUniversitat Politecnica de [email protected]

MS53

On the 2-Edge Connectivity of Geometric PlanarGraphs With Bounded Edge Length


Oscar Morales PonceCarleton UniversityOttawa, [email protected]

MS53

Some Results about Disk Graphs and PseudocircleArrangements


Tobias MuellerCentrum voor Wiskunde en [email protected]

MS54

Generating Trees for Partitions and Permutationswith No k-nestings

We describe a generating tree approach to the enumera-tion and exhaustive generation of set partitions and per-mutations with no k-nestings. Unlike previous work in theliterature using connections of these objects with Youngtableaux and restricted lattice walks, our approach dealsdirectly with partition and permutation diagrams. We de-tail a generation algorithm that uses only labels to describethese set partitions and permutations with no k-nestings,and provide explicit functional equations for the generat-ing functions, with k as a parameter, getting valuable seriesresults.

Sophie Burrill

96 DM12 Abstracts

Simon Fraser [email protected]

Sergi ElizaldeDartmouth [email protected]

Marni MishnaSimon Fraser [email protected]

Lily YenCapilano UniversitySimon Fraser [email protected]

MS54

Bijections Between Truncated Affine Arrange-ments and Valued Graphs

We present some contructions on the set of nbcs (No Bro-ken Circuit sets) of some deformations of the braid arrange-ment. This leads us to some new bijective proofs for Shi,Linial and similar hyperplane arrangements.

David ForgeUniversite de [email protected]

veronique ventosUniversite Paris [email protected]

sylvie corteelUniversite Paris [email protected]

MS54

What Computer Algebra Systems Can Offer toTackle Realizability Problems of Matroids

In his PhD dissertation, Hiroki Nakayama establishes aconnection between oriented matroids and mathematicalprogramming, by applying polynomial optimization tech-niques to the realizability problem of oriented matroids. Inthis talk, I will discuss what are the tools that computeralgebra systems offer today to implement realizability al-gorithms, such as the one of Nakayama.


MS54

Stochastic Ptri Nets and Symbolic Calculus


Michel PetitotUniversite des Sciences et Technologies de Lille [email protected]

MS54

The Interplay Between Finite Fields and Computer

Algebra

Finite fields display erratic behavior when the field size issmall and for large finite fields memory usage and processorpower become bottlenecks. I will outline the computer al-gebra packages containing finite fields implementations andwill list drawbacks and advantages to each. I will illustratethis with examples from my research including an exhaus-tive search algorithm for normal bases as well as data setsobtained for a section of the forthcoming ”Handbook ofFinite Fields”.

David ThomsonCarleton [email protected]

MS55

A Theoretical Foundation for Neural Sensor Net-works

Multiple recent multi-neuron recordings have uncoveredstructure in the statistics of mammalian retinal outputthat is well-captured by a probabilistic model from sta-tistical physics called the Ising model. Guided by thesefindings and motivated by classical ideas in theoretical neu-roscience, we have developed a model of neural sensor net-works. Our approach incorporates a recently developedIsing model parameter fitting technique called MinimumProbability Flow (MPF). Although a theoretical explana-tion for network processing of sensory stimuli, our modelcan also be used to uncover structure in many highly noisydatasets. We discuss several such applications to neuro-science, behavioral genetics, and machine learning. In ad-dition, we discuss how our network can learn an exponen-tial number of combinatorial structures (cliques in graphs)from only a small number of training samples.

Christopher HillarUniversity of California, [email protected]

Kilian KoepsellUC [email protected]

MS55

Neural Code Topology Imposes Constraints on theNetwork Architecture

There are two complementary perspectives on what con-trols neural activity in sensory systems: receptive fields,and neural network dynamics. Receptive fields largely de-termine the representational function of a neuronal net-work, while network dynamics stem from the networkstructure. We propose a mathematical framework wherethe structure of neuronal networks can be related to com-binatorial properties of receptive fields, and demonstratehow this can be used to relate network structure to homo-topy invariants of represented stimuli.

Vladimir ItskovDepartment of MathematicsUniversity of Nebraska, [email protected]

MS55

Stable Exponential Storage in Hopfield Networks

A Hopfield auto-associative neural network on n nodes can

DM12 Abstracts 97

store a random collection of at most 2n binary patterns asmemories. However, it has been observed experimentallythat random networks are typically capable of exponentialstorage in the form of spurious memories, which emergemysteriously from the random network coupling. We givefor the first time a parameterized family of Hopfield net-works that stably store exponential numbers of binary pat-terns. Moreover, we give experimental evidence suggestingthat these memories are attractors for the clean-up dynam-ics of the Hopfield network with large basins of attraction.We also present some applications.

Kilian KoepsellUC [email protected]

Ngoc Tran, Christopher HillarUniversity of California, [email protected], [email protected]

MS55

Noise-Induced Dynamics in Electricaly CoupledNeuronal Networks

Electrically coupled networks of excitable neurons forcedby small noise are analyzed in this work, using techniquesfrom dynamical systems and algebraic graph theories. Theresults are presented from two complementary perspectivesof variational analysis of spontaneous network dynamicsand the slow-fast analysis of synchronization. The for-mer yields geometric interpretation of various dynamicalregimes, while the latter highlights the contribution of thenetwork architecture to the stability of the synchronousstate.

Georgi S. MedvedevDrexel UniversityDepartment of [email protected]

MS55

Simplex Packings of Marginal Polytopes and Mix-tures of Exponential Families

A combinatorial approach to the representational powerof mixtures of discrete exponential families is presentedbased on combinatorics of convex polytopes and codingtheory. The number of mixtures of a hierarchical modelwhich contain another hierarchical model can be boundedby the cardinality of simplex face packings of their marginalpolytope lattices. A bound for such packings is presentedfor the k-way interaction exponential families.

Guido F. MontufarPennsylvania State [email protected]

40 dm12 abstracts ip0 dnes knig prize lecture: talktitle tbd...

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