dm08 abstracts - society for industrial and applied ... · dm08 abstracts 29 michael o. albertson...

28 DM08 Abstracts

CP1

A Characterization of the Tree-Decomposition of 3-Trees That Can Be Represented as Unit RectangleVisibility Graphs

Building on work of Dean, Ellis-Monaghan, Hamiltonand Pangborn, we study unit rectangle visibility graphs(URVGs), that is, graphs whose vertices can be representedby axis-aligned unit squares in the plane and whose adja-cencies are represented by vertical or horizontal visibilitybetween squares. The former work characterizes trees thatare URVGs. We characterize the tree-decompositions of 3-trees that are URVGs as those whose tree decompositionsbelong to a class of trees closely related to caterpillars ofmaximum degree 4.

Joan P. HutchinsonMacalester CollegeDepartment of [email protected]

Alice M. DeanSkidmore [email protected]

CP1

Minkowski Fonctionals on a Hexagonal Tesselation

Any pattern in a cubic tessellation can be decomposedinto open elements (vertices, edges, faces, cubes...). Thefunctionals Fl which count the number of elements Ol areadditive, motion invariant and monotonically increasing.Any additive and motion-invariant functional must be alinear combination of the Fl. It is already known that Fl

can be used to determine the typical geometric quantities.We have shown that this approach can be extended to anhexagonal tessellation.

Isabelle BlasquezLimoges [email protected]

Pierre fournierLICN-IUT du [email protected]

Jean-Franois PoiraudeauLimoges University, [email protected]

CP1

Convex Subdivisions with Low Stabbing Numbers

It is shown that for every subdivision of the d-dimensionalEuclidean space, d ≥ 2, into n convex cells, there is astraight line that stabs at least Ω((log n/ log log n)1/(d−1))cells. In other words, if a convex subdivision of d-space hasthe property that any line stabs at most k convex cells, thenthe subdivision consists of at most exp(O(kd−1 log k)) cells.This bound is best possible apart from a constant factor.It was previously known only in the case d = 2.

Csaba D. TothUniversity of [email protected]

CP1

Large Quadrant - Depth

Let P be a set of n points in Rd. We make an attemptto characterize the set of all pairs (α, β) such that therealways exists a point z and two opposite orthants, deter-mined by the axes parallel hyperplanes through z, suchthat one contains at least αn points of P and the other atleast βn points of P. We explore both the case where z isrequired to belong to the set P and also the general casewhere z may be any point in Rd. Some generalizations ofthis problem are also considered.

Itay Ben-DanMath Dept. TechnionHaifa 32000 [email protected]

Stefan FelsnerTechnical University of BerlinInstitute for [email protected]

Rom PinchasiMath Dept. TechnionHaifa 32000 [email protected]

Ran ZivTel-Hai Academic CollegeUpper Galilee, [email protected]

CP2

The A4-Structure of a Graph

In analogy with the P4-structure of a graph, we definethe A4-structure of a graph G to be the 4-uniform hy-pergraph on V (G) whose edges are vertex subsets inducingalternating 4-cycles. We present several analogues for A4-structure of known results on P4-structures, and in doingso we provide further motivation for the work of R. Tyshke-vich on canonical decomposition of a graph [Decompositionof graphical sequences and unigraphs. Discrete Math. 220(2000), no. 1-3, 201–238].

Michael D. BarrusUniversity of Illinois at [email protected]

Douglas B. WestUniv of Illinois - UrbanaDepartment of [email protected]

CP2

Grunbaum Colorings of Toroidal Triangulations

Grunbaum’s Conjecture, that every triangulation of an ori-entable surface has a 3-edge-coloring such that each facialtriangle receives three colors, was recently disproven forsurfaces of genus 5 and higher by Kochol. In contrast, weshow that the conjecture holds for toroidal triangulationsthat do not have chromatic number 5.

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

DM08 Abstracts 29

Michael O. AlbertsonSmith [email protected]

Hannah AlpertUniversity of [email protected]

Ruth HaasSmith [email protected]

CP2

The Kuratowski Covering Conjecture for Graphsof Order < 10

Kuratowski proved that a finite graph embeds in the planeif it does not contain a subdivision of either K5 or K3,3,called Kuratowski subgraphs. A generalization of this re-sult to all nonorientable surfaces says that a finite graphembeds in the nonorientable surface of genus g if it does notcontain g + 1 Kuratowski subgraphs such that the unionof each pair of these fails to embed in the projective plane,the union of each triple of these fails to embed in the Kleinbottle if g ≥ 2, and the union of each triple of these failsto embed in the torus if g ≥ 3. We prove this conjecturefor all graphs of order < 10.

Suhkjin HurDepartment of MathematicsThe Ohio State [email protected]

CP3

Totally Greedy Coinsets and Greedy Obstructions

A coin set C = (a1, a2, . . . , ak) is a list of positive inte-gers with ak > ak−1 > . . . > a2 > a1, where a1 is alwaysset to be 1. Pearson has provided a polynomial time al-gorithm for determining whether a coinset is greedy, thatis when a simple greedy change-making algorithm that al-ways chooses the largest denomination coins possible, willalso produce the fewest number of coins in change. We con-sider a stricter properties on coin sets, called total greed-iness, which requires that all initial subsequences of thecoinset also be greedy, and present a simple property thatmakes it easy to test if a coinset is totally greedy. Finally,we begin to explore the theory of greedy obstructions–those coinsets that cannot be extended to greedy coinsetsby the addition of coins in larger denominations.

Robert CowenQueens College, [email protected]

Lenore CowenDept. of Computer ScienceTufts [email protected]

Arthur SteinbergQueens College, [email protected]

CP3

A Discrete Optimization Formulation for the Min-

imum Cost Vaccine Formulary Selection Problem

As the complexity of the United States RecommendedChildhood Immunization Schedule increases, a combina-torial explosion of choices is being presented to publichealth policy makers and pediatricians. A discrete op-timization problem, termed the General Minimum CostVaccine Formulary Selection Problem (GMCVFSP), is pre-sented, which models a general childhood immunizationschedule. Exact algorithms and heuristics for GMCVFSPare discussed. Computational results are also reported.The results reported provide fundamental insights into thestructure of the GMCVFSP model.

Shane [email protected]

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

Edward SewellSouthern Illinois University [email protected]

CP3

A Min-Max Theorem on a Class of BipartiteGraphs with Applications

We prove a theorem on the class of bipartite graphs that donot contain any induced cycles on exactly six vertices. Us-ing this, we show that the size of a largest induced match-ing equals the size of a smallest cover with chain subgraphsfor the class of chordal bipartite graphs. Further, we showthat a largest induced matching and a smallest cover withchain subgraphs can be found in polynomial time for theclass of chordal bipartite graphs, and also present efficientalgorithms for a subclass of chordal bipartite graphs. Theseproblems are NP-hard for general bipartite graphs and ourwork generalizes previously known results.

R. Sritharan, Atif Abueida, Arthur BuschThe University of [email protected],[email protected],[email protected]

CP4

Non-Cover Generalized Mycielski, Kneser, andSchrijver Graphs

A graph is said to be a cover graph if it is the underlyinggraph of the Hasse diagram of a finite partially ordered set.We prove that the gereralized Mycielski graphs of an oddcycle, Kneser graphs KG(n, k), and Schrijver graphs SG(n,k) are not cover graphs when n ≥ 2k + 2.

Chen-Ying LinDepartment of Computer Science and InformationEngineering,Shu-Te University, Kaohsiung 824, [email protected]

Ko-Wei LihInstitute of Mathematics, Academia Sinica, [email protected]

30 DM08 Abstracts

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

CP4

On the Multi-Dimensional Frobenius Problem

We consider the problem of finding maximal (in an appro-priate partial order) solutions g to the linear Diophantinesystem Mx = g. We extend past work and prove severalreduction formulas. This is work done by undergraduatesin an REU program.

Vadim PonomarenkoSan Diego State UniversityDepartment of Mathematics and [email protected]

Jeffrey AmosKansas State [email protected]

Charles ChenHarvard [email protected]

Olga LepiginaColorado [email protected]

Timur NezhmetdinovLehigh [email protected]

Darren OngTexas Christian [email protected]

Matthew RicherArizona State [email protected]

Laura ZirbelLoyola Marymount [email protected]

CP4

Fractional Weak Discrepancy, Interval Orders, andForbidden Configurations

The fractional weak discrepancy wdF (P ) of a poset P =(V,≺) is the minimum nonnegative k for which there ex-ists a function f : V → R satisfying (i) if a ≺ b thenf(a) + 1 ≤ f(b) and (ii) if a ‖ b then |f(a)− f(b)| ≤ k. Anr + s is a disjoint union of two chains with r and s elements,respectively. Semiorders, which contain no induced 2 + 2or 3 + 1, were characterized by their fractional weak dis-crepancy in Shuchat, Shull, and Trenk, ORDER, 23:51–63,2006. Here we generalize this result to describe the rangeof values of wdF (P ) based on whether or not P containscertain induced r + s configurations. For example, we findthe range of wdF (P ) for interval orders with no inducedn + 1.

Alan ShuchatMathematics DepartmentWellesley [email protected]

Randy ShullDepartment of Computer ScienceWellesley [email protected]

Ann N. TrenkWellesley [email protected]

CP4

The Fractional Weak Discrepancy of SplitSemiorders

The fractional weak discrepancy wdF (P ) of a poset P =(V,≺) is the minimum nonnegative k for which there existsa function f : V → R satisfying (i) if a ≺ b then f(a) +1 ≤ f(b) and (ii) if a ‖ b then |f(a) − f(b)| ≤ k. In thistalk we extend previous results on the range of wdF (P ) forsemiorders to split semiorders (also known as unit point-core bitolerance orders). In particular, we prove that forsuch posets the range is the set of rationals r/s for whichs > 0 and s− 1 ≤ r < 2s.

Randy ShullDepartment of Computer ScienceWellesley [email protected]

Alan ShuchatMathematics DepartmentWellesley [email protected]


CP5

A Generalization of the Laplacian CharacteristicPolynomial

A generalization of the characteristic polynomial of theLaplacian matrix of a simple graph is proposed, from whichthe Laplacian characteristic poly for the graph as well asfor its complement can be derived as special cases. Also,graphs with the same generalised characteristic polynomialare discussed.

Biswajit DebSMIT, SIKKIM, INDIAPin. 737132biswajit [email protected]

Sukanta DasSMIT, SIKKIM, [email protected]

CP5

On Middle Cube Graphs

The middle cube graph of the cube Q2k+1 contains verticeswhose binary representation has either

(2k+1

k

)or

(2k+1k+1

)

number of 1’s and their associated edges. This family ofgraphs has been proposed as model of interconnection net-work because of its properties. The middle cube graphcan be obtained from the well-known odd graphs. Here westudy some of their properties under the light of the theory

DM08 Abstracts 31

of distance regular graphs.

Lali Barriere, Cristina Dalfo, Miquel ngel FiolDepart. Matematica Aplicada IVUniversitat Politecnica de [email protected], [email protected], [email protected]

Margarida MitjanaDepart. Matematica Aplicada IUniversitat Politecnica de [email protected]

CP5

Equivalences on Graph Dynamical Systems (GDS)

GDS are discrete dynamical systems over graphs Y . Forasynchronous GDS the dynamical system map is specifiedusing a permutation of the vertices of Y . Building on func-tional equivalence of asynchronous GDS, which is charac-terized through the acyclic orientations of Y , we presentnew results showing how orbit equivalence of GDS is cap-tured by group actions on acyclic orientations, and how theequivalence classes are enumerated through the Mobius in-variant of Y . The connection to Coxeter theory will beshown.

Henning S. Mortveit

Assistant Professor/Senior Research AssociateVirginia [email protected]

CP5

Spectral Classification of Regular Graphs andHamiltonian Cycles

We present new results on the connection between the de-terminant of a matrix function of subgraphs and the longestcycles in a given graph. We show that certain functionalsdependent on graph spectrum exhibit a fractal-like, mul-tifilar structure, and explain this phenomenon using theIhara-Selberg formula and geodesics. This structure sepa-rates non-Hamiltonian regular graphs into bridge and hardcamouflaged graphs. A new recursive formula for the car-dinality of regular bridge graphs will also be presented.

Giang T. NguyenUniversity of South [email protected]

CP5

Subgraphs of Singular Graphs

Characterization of singular graphs can be reduced to thenon-trivial solutions of a system of linear homogeneousequations Ax = 0 for the 0-1 adjacency matrix A. A graphG is singular of nullity η(G) ≥ 1, if the dimension of thenullspace ker(A) of its adjacency matrix A is η(G) . Wedetermine necessary and sufficient conditions for a graphto be singular in terms of admissible induced subgraphs.

Irene N. ScirihaUniversity of [email protected]

CP6

The Coloring Graph

Two colorings of a graph, G, are isomorphic if by permuting

the colors in one of them, we can obtain the other. The setof nonisomorphic colorings of G is the set of isomorphismclasses of proper colorings. Define the graph of nonisomor-phic colorings of G, I(G), to have vertex set equal the setof nonisomorphic colorings of G, with an edge between twocolorings if they are isomorphic on V(G-x for some x inV(G). In this talk we explore properties of I(G).

Ruth HaasSmith [email protected]

Ina PetkovaColumbia [email protected]

CP6

Bounding the Chromatic Number of Claw-FreeGraphs

Reed conjectured that for any graph, χ ≤ � 12(Δ + 1 + ω)�,

where χ, Δ, and ω are the chromatic number, maximumdegree, and clique number respectively. This conjecturehas recently been proven for line graphs and quasi-linegraphs. We extend these results, proving the conjecture forall claw-free graphs and discussing a stronger (and some-times more manageable) version of the conjecture.

Andrew D. King, Bruce ReedMcGill [email protected], [email protected]

CP6

Graph Partitions

Let H be a graph with vertex set V (H) = {1, 2, . . . , n},and let C1, C2, . . . , Cn be sets of graphs. An(H;C1, C2, . . . , Cn)-partition of a graph G is an orderedpartition (V1, V2, . . . , Vn) of V (G) such that, for each i,the subgraph of G induced by Vi belongs to Ci and, fur-ther, there is an edge joining a vertex in Vi to a vertexin Vj only if ij is in E(H). When each set Ci equals

{Kn : n = 0, 1, 2, . . .} this concept coincides with theusual notion of homomorphism of G to H. If, in addi-tion, H is complete, it coincides with the usual notionof n-colouring of G. We describe polynomial time algo-

rithms for the case where H is triangle-free and each setCi is a ”starter matrix class” of graphs – a large collectionof graphs classes that includes cliques, stars, complete k-partite graphs, and graphs with a dominating vertex. Wealso give NP-completeness results in a variety of cases whenH contains a triangle, for example when each set Ci is theset of k-partite graphs.

Gary MacgillivrayMathematics and StatisticsUniversity of [email protected]

Peter DukesUniversity of Victoria, [email protected]

Steve LowdonMathematics and StatisticsUniversity of [email protected]

32 DM08 Abstracts

CP6

Complexity of Generalised Colourings of ChordalGraphs

In our contribution, we investigate generalised colouringproblems in the class of chordal graphs. In particular, wefocus on colouring problems in which the colour classes areavoiding particular types of forbidden induced subgraphs.These include for instance problems like split colourings,polar colourings, subcolourings and many others. We de-scribe recent complexity results for cases with small for-bidden induced subgraphs and outline possible extensionsof these results.

Juraj StachoSimon Fraser [email protected]

CP7

Graphs without a C4 or a Diamond

In general, it is not clear how to generalize results onchordal graphs to C4-free graphs and there are a number ofopen questions regarding C4-free graphs. We consider theclass of (C4, diamond)-free graphs. We provide an efficientrecognition algorithm, and count the number of maximalcliques and the number of n-vertex labeled graphs. Wealso give an efficient algorithm for finding a largest cliquein the more general class of (house, diamond)-free graphs.

Elaine M. EschenLane Department of CSEE,West Virginia [email protected]

Chinh HoangWilfrid Laurier UniversityDept of Physics and Computing [email protected]

Jeremy SpinradVanderbilt [email protected]

R. SritharanUniversity of [email protected]

CP7

Simultaneously Chordal Graphs

Two graphs, G1 and G2, sharing some vertices S, are si-multaneously chordal if we can add edges E′ between G1−Sand G2 −S to create a chordal graph G1 ∪G2∪E′. Equiv-alently, G1 and G2 have an intersection representation assubtrees of one common tree. This concept is related to si-multaneous planar embeddings and to the graph sandwichproblem. In contrast with the graph sandwich problems,graph simultaneity problems are in P for many interestingproperties. We give an efficient algorithm for simultaneouschordality.

Krishnam Raju Jampani, Anna LubiwUniversity of [email protected], [email protected]

CP7

How Vertex Elimination Can Overachieve

Simplicial (‘perfect) elimination orderings of chordalgraphs enjoy many advantages that overlap and are eas-ily confused—for instance, the total unimportance of thechoices among simplicial vertices (removing the schemingfrom elimination schemes). What assures this bonus when‘simplicial is replaced with an arbitrary vertex propertyφ(v)? Or, what guarantees the existence of choices at ev-ery step in φ-elimination? The graph metatheory of vertexelimination involves suitable distinctions, linked by simpletheorems and delineating counterexamples.

Terry McKeeWright State UniversityDepartment of Mathematics & [email protected]

CP7

Local Steiner Convexity in Graphs

Let G be a connected graph and S ⊆ V (G). A Steinertree T for S is a minimum subtree of G with S ⊆ V (T ).The Steiner distance of S is |E(T )|. A set S ⊆ V (G) isgk-convex, if for every k-subset R ⊆ S, every Steiner treeT for R satisfies V (T ) ⊆ S. Considering three local g3-convexities, we characterize graphs satisfying either of thefirst two and determine some necessary and some sufficientconditions for a graph to satisfy the third.

Morten H. NielsenUniversity of [email protected]

Ortrud R. OellermannThe University of [email protected]

Michael A. HenningUniversity of [email protected]

CP8

Some Non-Simple Forbidden Configurations andDesign Theory

Let k, l,m, q be given. We let f(m, k, l, q) denote the max-imum number of subsets of {1, 2, . . . ,m} in a family F sothat we cannot find q sets A1, A2, . . . , Aq ∈ F and k + lelements a1, a2, ..., ak+l ∈ {1, 2, . . . ,m} so that each of theq sets Ai contains the k elements a1, a2, . . ., ak and doesnot contain the l elements ak+1, ak+2, . . ., ak+l. We areable to compute f(m, 1, 1, q), f(m, 2, 1, q) and f(m, 2, 2, q)exactly for large m. For example, the natural construc-tion for k = 2, l = 1 is to take all sets of size 0, 1, 2, mand also the sets of size 3 corresponding to a simple triplesystem of λ = q − 2 (which exists by a result of Dehon,1983) and we are able to show for large m that indeed

f(m, 2, 1, q) =(m0

)+(m1

)+(m2

)+ q−2

3

(m2

)+(mm

).

Richard P. AnsteeMathematics DepartmentUniversity of British [email protected]

Farzin BarekatMathematics Deaprtment, UBC

DM08 Abstracts 33

farzin [email protected]

CP8

About Convexity of One Function from CodingTheory

We complete the proof of upper bound for the multiplepacking in q−ary Hamming space.

Vladimir BlinovskyInstitute of Information Transmission [email protected]

CP8

Avoiding Partial Latin Squares

Let P be an n× n array of symbols. P is called avoidableif for every set of n symbols there is an n×n Latin squareL on these symbols so that corresponding cells in L andP differ. We give a short argument that shows all partialLatin squares of order at least 4 are avoidable. We alsoshow, if time permits, that a partial generalized sudokusquare of order n2 can be avoided by a generalized sudokusquare of order n2.

Jaromy S. KuhlUniversity of West [email protected]

Tristan DenleyUniversity of [email protected]

CP8

Perfect T (G) Triple Systems when G is a Matching

A T (G) triple is formed by taking a graph G and replac-ing every edge with a 3-cycle, where all of the new verticesare distinct from all others in G. An edge-disjoint decom-position of 3Kn into T (G) triples is called a T (G) triplesystem. Finally, suppose that G partitions Kn. If we candecompose Kn into copies of the graph G, where we canform a T (G) triple from each graph in the decompositionand produce a partition of 3Kn, then the resulting T (G)triple system is called perfect. The set of positive integersn, for which a decomposition of 3Kn into a perfect T (G)triple system exists is called the spectrum. The spectrafor all graphs on 4 or fewer vertices have been completelydetermined by Billington, Kucukcifci, Lindner, and Rosa.However, the spectra for graphs on arbitrary numbers ofvertices is still an open problem. In this talk, I will discussjoint work with Danny Dyer on determining the spectrumwhen G is an arbitrary matching. In particular I will iden-tify infinite families for which we have determined the spec-tra and illustrate a relationship with generalized extendedSkolem sequences of multiplicity 3.

Josh ManzerMemorial University of [email protected]

CP8

Combinatorial Constructions for Optimal Two-Dimensional Optical Orthogonal Code

Optical orthogonal codes (OOCs) have been designed forOCDMA. An one dimensional (1-D) optical orthogonalcode (1-D OOC) is a set of one-dimensional binary se-

quences having good auto and cross-correlations. One lim-itation of 1-D OOC is that the length of the sequence in-creases rapidly when the number of users or the weight ofthe code is increased, which means large bandwidth ex-pansion is required if a big number of codewords is needed.To lessen this problem, two-dimensional (2-D) coding (alsocalled multiwavelength OOCs) was invested. An two di-mensional (2-D) optical orthogonal code (2-D OOC) is a setof u× v matrix with (0, 1) elements having good auto andcross-correlations. Recently many researchers are workingon constructions and designs of 2-D OOCs. In this pa-per, we shall reveal the combinatorial properties of 2-DOOCs and give an equivalent combinatorial description of2-D OOC. Based on this, we are able to use combinatorialmethods to obtain many optimal 2-D OOCs.

Ruizhong WeiDepartment of Computer ScienceLakehead [email protected]

CP9

Strong Cayley Graphs, Weak Cayley Graphs andNon-Cayley Graphs

In this talk, I will first discuss strong Caley graphs andweak Cayley graphs. I will then present examples of suchgraphs, as well as non-Cayley graphs.

Yuqing ChenDepartment of Mathematics and StatisticsWright State [email protected]

CP9

Independent Domination Critical and BicriticalGraphs

A graph is independent domination critical if the removal ofany vertex decreases the independent domination numberand it is independent domination bicritical if the removalof any two vertices reduces the independent dominationnumber. We present new results in this area, includingvarious construction methods and diameter results.

Michelle EdwardsUniversity of VictoriaDepartment of Mathematics and [email protected]

Arthur FinbowSaint Mary’s UniversityDepartment of Mathematics and Computing [email protected]

Gary MacGillivrayUniversity of VictoriaDepartment of Mathematics and [email protected]

CP9

Graceful Directed Graphs

A graph G consists of a set of vertices and a set of edges.If a nonnegative integer f(v) is assigned to each vertex vof G then the vertices of G are said to be labeled. G isitself a labeled graph if each edge e=uv is given the valuef(uv) = f(u)*f(v), where * is a binary operation. In lit-

34 DM08 Abstracts

erature one can find the binary operation * as addition,multiplication, modulo addition or absolute difference ormodulo subtraction. Graph labelings, where the verticesand edges are assigned, real values subject to certain con-ditions, have often been motivated by their utility to vari-ous applied fields and their intrinsic mathematical interest(logico mathematical). A directed graph D with n nodesand e arcs, no self-loops and multiple edges is labeled byassigning to each node a distinct element from the set Ze+1= 0,1,2,,e. An arc (x,y) from node x to y is labeled with?(xy)= (?(x)-?(y))(mod (e+1)), where ?(x) and ?(y) arethe values assigned to the nodes x and y. A labeling isa graceful labeling of D if all ?(xy) are distinct. Then Dis called a graceful digraph. In this talk we discuss aboutgraceful labelings of directed graphs and their applicationsto algebra ( sequenciable cyclic groups, complete mappingsand neofields).

Suresh M. Hegdenational Institute of Technology [email protected]

CP9

Triangle-Free Graphs and Greedoids

A matching is uniquely restricted if its saturated verticesinduce a subgraph having a unique perfect matching. S isa local maximum stable set of G, and we write S ∈ Ψ(G),if S is a maximum stable set of G[N [S]]. Nemhauser andTrotter Jr. proved that any S ∈ Ψ(G) is a subset of amaximum stable set of G. We demonstrate that if G istriangle-free, then Ψ(G) is a greedoid iff all its maximummatchings are uniquely restricted and G[N [S]] is a Konig-Egervary graph for any S ∈ Ψ(G).

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]

CP9

Labeling Graphs with Three Levels of Constraints

Motivated by the channel assignment problem, a graph la-beling with conditions that depend on the distance betweenvertices (called distance labeling) has been proposed andstudied by many authors. In previous studies, these dis-tance labelings with two constraints have been considered.In this talk, we will consider distance labelings with threeconstrains. In particular, given a graph G = (V,E) andpositive integers d1, d2, d3, an L(d1, d2, d3)-labeling is anassignment f : V → {0, 1, · · ·} such that |f(u)− f(v)| ≥ diwhenever the distance vertices u and v is i in G, fori = 1, 2, 3. We seek a smallest integer k (denoted byλ(G; d1, d2, d3) so that there is an L(d1, d2, d3)-labeling onG with the maximum value k. We study this value onseveral classes of graphs.

Roger YehDept. of Applied Math., Feng Chia [email protected]

Honya Liao, Cian-Hui YangDept. of Applied Math., Feng Chia [email protected], [email protected]

CP10

Subdivided Graphs As Isometric Subgraphs ofHamming Graphs

Isometric subgraphs of Hamming graphs (resp. hyper-cubes) are called partial Hamming graphs (resp. partialcubes). Partial cubes have first been investigated in [Gra-ham Pollack, On the addressing problem for loop switch-ing,1971] and [Djokovic, Distance preserving subgraphs ofthe hypercubes,1973]. In this talk we present structuralcharacterizations of partial Hamming graphs. It is proventhat given G a subdivision of a clique Kn (n ≥ 1), G isisometrically embeddable in a Hamming graph if and onlyif G is a partial cube or G = Kn. The characterization forsubdivided wheels is also obtained.

Laurent BeaudouInstitut [email protected]

Sylvain GravierCNRS – Institut FourierERTE Maths a [email protected]

Kahina MeslemLAID3 – USTHBInstitut Fourierkahina [email protected]

CP10

A Formula for the Kirchhoff Index

We show here that the Kirchhoff index of a network is theaverage of the Wiener capacities of its vertices. Moreover,we obtain a closed-form formula for the effective resistancebetween any pair of vertices when the considered networkhas some symmetries which allows us to give the corre-sponding formulas for the Kirchhoff index. In addition, wefind the expression for the Foster’s n-th Formula.

Angeles Carmona, Enrique Bendito, Andres Encinas, JoseGestoDepartment of Applied Mathematics IIIUniversitat Politecnica de [email protected], [email protected], [email protected], [email protected]

CP10

Pizza Delivery: 2-Stop-Return Distances in Graphs

A delivery person must leave a central location, deliverpackages at various addresses, and return by the mostefficent route. This motivates the following definition.Given a set of k + 1 distinct vertices v, v1, v2, . . . , vk ina simple graph G, the k-stop-return distance from v to{v1, v2, . . . , vk} is defined to be

dks(v, v1, v2, . . . , vk) = minP(S) (d(v, v1) + d(v1, v2) + . . . + d(vk, v)) ,

where P(S) is the set of all permutations of {v1, v2, . . . , vk}.We consider some questions related to the 2-stop-returnradius, diameter, center, and periphery.

Grady Bullington, Linda Eroh

DM08 Abstracts 35

University of Wisconsin, [email protected], [email protected]

Ralucca M. GeraNaval Postgraduate [email protected]

Steven J. WintersUniversity of Wisconsin [email protected]

CP10

Minimum Size of a Graph of Given Diameter

We determine the minimum size of a graph of fixed diam-eter and fixed minimum degree. This answers a questionof J.A. Bondy and U.S.R. Murty, Extremal Graphs of di-ameter two with prescribed minimum degree, Studia Sci.Math. Hungar. 7 (1972), 239-241. We relate this result toa problem in graph pebbling.

Peter DankelmannUniversity of KwaZulu-NatalDurban, South [email protected]

John SchmittMiddlebury CollegeMiddlebury, [email protected]

Lutz VolkmannLehrstuhl II fur MathematikAachen, [email protected]

CP10

The (a,b)-Forcing Geodetic Graphs

For every pair of vertices u, v in a graph, a u-v geodesic isa shortest path from u to v. For a graph G, let IG[u, v]denote the set of all vertices lying on a u-v geodesic, andfor S ⊆ V (G), let IG[S] denote the union of all IG[u, v] forall u, v ∈ S. A set S ⊆ V (G) is a geodetic set if IG[S] =V (G). The geodetic number g(G) of a graph G is theminimum cardinality of a geodetic set in G. A subset F ⊆V (G) is called a forcing subset of G if there exists a uniqueminimum geodetic set containing F . A forcing subset F iscritical if every proper subset of F is not a forcing subset.The cardinality of a minimum critical forcing subset in Gis called the forcing geodetic number f(G) of G and thecardinality of a maximum critical forcing subset in G iscalled the upper forcing geodetic number f+(G) of G. If Gis a graph with f(G) = 0, then G has a unique minimumgeodetic set; that is, f+(G) = 0. In the paper, we provethat, for any nonnegative integers a, b and c with 1 ≤ a ≤b ≤ c−2 or 4 ≤ a+2 ≤ b ≤ c, there exists a connected graphG with f(G) = a, f+(G) = b, and g(G) = c. This resultsolves a problem of Zhang in [The upper forcing geodeticnumber of a graph, Ars Combinatorica 62 (2002), 3-15].

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

CP11

Periodicity and Other Structure in a Colorful Fam-

ily of Nim-like Arrays

We study aspects of the combinatorial and graphical struc-ture shared by a certain family of recursively generatedarrays related to the operation of Nim-addition. In partic-ular, these arrays display periodic behavior along rows anddiagonals. We explain how various features of computer-generated graphics depicting these arrays are reflections ofthe theorems we prove.

Lowell AbramsDepartment of MathematicsThe George Washington [email protected]

Dena MortonDepartment of Mathematics and Computer ScienceXavier [email protected]

CP11

Periods, Partial Words, and a Result of Guibas andOdlyzko

A well known and unexpected result of Guibas and Odlyzkostates that the set of all periods of a word is independentof the alphabet size. We provide an algorithm that givena nonspecial partial word u, or word with “do not know”symbols or “holes,” computes a binary partial word v shar-ing the same sets of periods and weak periods as u, andsatisfying H(u) ⊃ H(v) where H(u), H(v) denote the setsof holes of u, v.

Francine Blanchet-SadriUniversity of North [email protected]

Brian ShireyNC State UniversityDepartment of Computer [email protected]

CP11

Hsiao-Code Check Matrices and Recursively Bal-anced Matrices

The key step of generating the well-known Hsiao code isto construct a {0, 1}-check-matrix in which each columncontains the same odd-number of 1’s and each row containsthe same number of 1’s or differs at most by one for thenumber of 1’s. and no two columns are identical in thematrix. Here, we focus on how to practically generate thecheck matrix of Hsiao codes. We have modified the originalalgorithm to be more efficient and effective.

Li ChenUniversity of the District of [email protected]

CP12

Construction of Transformation and SummationFormulas Using Symbolic Operator Approach andTheir Applications

Here we present the construction of pairs of transforma-tion and expansion formulas by using a kind of symbolicoperator method and generalized Eulerian fractions. Theapplications of the formulas for various types of power se-

36 DM08 Abstracts

ries to the computational number theory are also discussed.

Tian-Xiao HeIllinois Wesleyan [email protected]

CP12

Minimal Tile and Bond-Edge Types for Self-Assembling DNA Graphs

We examine a model for self-assembling DNA graphicalcomplexes using tiles of branched DNA molecules with freesticky ends. We address determining the minimum numberof tile and bond edge types necessary to create a givengraph under three different scenarios: 1. Where the inci-dental creation of complexes of smaller size than the targetgraph is acceptable; 2. Where the incidental creation ofcomplexes the same size as the target graph is acceptable,but not smaller complexes; 3. Where no complexes thesame size or smaller than the target graph are acceptable.In each of these cases, we find bounds for the minimum andmaximum number of tile and edge types that must be de-signed and give specific minimum values for common graphclasses (complete, bipartite, trees, regular, etc.). For theseclasses of graphs, we provide either explicit descriptions ofthe set of tiles achieving the minimum number of tile andbond edge types, or efficient algorithms for generating thedesired set.

Greta PangbornSaint Michael’s [email protected]

Laura BeaudinSaint Michaels [email protected]

Joanna Ellis-MonaghanSaint Michaels CollegeDepartment of [email protected]

Natasha JonoskaUniversity of South FloridaDepartment of [email protected]

Dave MillerSaint Michaels [email protected]

CP12

The Solution of Combinatorial Problems UsingSAT

The satisfiability problem (SAT) has been very well ex-plored. Based on very efficient parallel algorithms for itssolution, it will be shown that many combinatorial prob-lems can be transformed into satisfiablity problems andsolved using these developed algorithms. The approach isconstructive and very general, no research procedures areinvolved, the results are always complete. It can be con-cluded that the solution algorithms for SAT can be used(in the sense of NP-completeness) for many other combi-natorial problems in a very general way.

Christian PosthoffUniversity of The West IndiesSt. Augustine Campus, Trinidad & Tobago

[email protected]

Bernd H. SteinbachFreiberg University of Mining and TechnologyInstitute of Computer [email protected]

CP12

Meanders and Stamp Foldings: Fast GenerationAlgorithms

Using a permutation representation, we present fast algo-rithms to exhaustively list all meanders and stamp fold-ings. The key to the algorithms is the introducion of adata structure that allows us to efficiently determine allpossible locations to insert the next meander crossing orstamp.

Joe SawadaUniversity of GuelphDept. of Computer [email protected]

CP13

On Classification of Rhotrices As Algebraic Struc-tures

In this paper, we presents the study of rhotrices and theirclassifications as algebraic structures of Groups, Semi-groups, Monoids, Ring,Field,Integral domain,Principalideal domain,Unique factorization domain and Bolean al-gebra. REFERENCES: [1] MOHAMMED, A.,(2007),Enrichment exercises through extension to rhotrices, In-ternational journal for Mathematical Education in Sci-ence and Techonlogy,38, 131-136. [2] AJIBADE, A. O.,(2003), The concept of rhotrix for mathematical enrich-ment.,International journal for Mathematical Education inScience and Techonlogy, 34,175-179. [3] SANI, B.,(2004),An alternative method for multiplication of rhotrices, In-ternational journal for Mathematical Education in Sci-ence and Techonlogy, 35, 777-781. [4] ATANASSOV, K.T.and SHANNON, A.G.(1998),Matrix-tertions and matrix-noitrets:exercises in mathematical enrichment, Interna-tional journal for Mathematical Education in Science andTechonlogy, 29, 898-903

Abdul MohammedDepartment of Mathematics, Ahmadu Bello University,Zaria, [email protected]

CP13

Recovery of Symmetry From Ghosts

If S is a finite subset of a vector space U and φ : U →V is an orthogonal projection onto a subspace V , thenφ(S) has “ghost symmetry” if the linear automorphismsof S ⊂ U may be recovered from the automorphisms ofφ(S) and its projections. We will see some examples andconsider the computational problems involved in studyingthis phenomenon in general

David RichterWestern Michigan [email protected]

DM08 Abstracts 37

CP13

Noncommutative Analogs of Monomial SymmetricFunctions, Cauchy Identity and Hall Scalar Prod-uct.

This talk introduces noncommutative analogs of mono-mial symmetric functions and fundamental noncommuta-tive symmetric functions. The expansion of ribbon Schurfunctions in both of these basis is nonnegative. With thesefunctions at hand, one obtains a noncommutative Cauchyidentity and can study a noncommutative scalar productimplied by Cauchy identity. This scalar product seems bea correct noncommutative analog of Hall scalar product inthe commutative theory.

Lenny TevlinYeshiva University, New [email protected]

CP14

Binary Subtrees with Few Path Labels

Let T be the rooted, perfect ternary tree of depth n. If eachedge in T is assigned a label in {0, 1}, then reading the la-bels along the edges in a path from the root of T to a leafin T induces a path label in {0, 1}n. Let f(T ) be the min-imum, over all perfect binary subtrees S of depth n of T ,of the number of path labels that occur in S. Let f(n) bethe maximum, over all {0, 1}-edge-labeled perfect ternarytrees T with depth n, of f(T ). The problem of boundingf(n) arose in an attack on a problem in computability the-ory, and seems to be combinatorially interesting in its ownright. We show that f(n) is at most (c/n)2n for some con-stant c, and that f(n) is at least (1.505)n for sufficientlylarge n. We conclude with some open problems.

Kevin MilansUniversity of [email protected]

CP14

Cut-Edges, 1-Blocks, and Matchings in RegularGraphs of Odd Degree

A block is a 1-block if exactly one edge joins it to the restof the graph. For a graph G, let b(G) be the number of 1-blocks, let c(G) be the number of cut-edges, and let α′(G)be the maximum size of a matching. Let F be the familyof (2r + 1)-regular graphs with n vertices. We prove that

b(G) ≤ (2r−1)n+2

4r2+4r−2and c(G) ≤ 2rb(G)−2r−1

2r−1for G ∈ F ,

which yields c(G) ≤ rn−(2r2+4r+1)

2r2+2r−1. We also prove that

α′(G) ≥ n2− rb(G)

2r+1, simplifying the recent proof by Henning

and Yeo that α′(G) ≥ n2− r

2(2r−1)n−1

(2r+1)(2r2+2r−1). For large r,

this bound is about n2− n

4r. All the bounds are sharp,

holding with equality for infinitely many n.

Suil OUniversity of Illinois, [email protected]


CP14

Bounds for the Real Number Labellings and La-bellings of the Triangular Lattice

The real number graph labelling is a general model of thechannel assignment problem. The transmitters are rep-resented by vertices and close transmitters are joined byedges. There are k types of edges and the level of interfer-ence is determined by non-negative parameters p1, ..., pk.The assignment of channels is represented by assigningnon-negative integers to vertices in such a way that the la-bels of vertices joined by an edge of the i-th type differ byat least pi. This model generalizes L(p1, ..., pk)-labellingsof graphs when the labels of vertices at distance exactlyd are required to differ by at least pd. The span of a la-belling is the maximal assigned label. The goal is to find alabelling with minimal span. We establish several generallower and upper bounds for real number graph labelingsand apply these bounds to L(p, q)-labelings of the infinitetriangular plane lattice. In particular, we determine theminimal spans of L(p, q)-labellings of the infinite triangu-lar plane lattice for all values of p and q (solving an openproblem of Griggs).

Petr SkodaCharles University, Czech [email protected]

Daniel KralCharles [email protected]

CP14

On Degenerate Colorings of Graphs on Surfaces

An r-coloring of G is degenerate if for every k ≤ r, theunion of any k color classes induces a (k − 1)-degenerategraph. This is a strenghtening of the notion of an acycliccoloring. We prove that every planar graph admits a degen-erate 11-coloring and that every graph of maximum degreeΔ has a degenerate coloring with O(Δ3/2) colors. The lat-ter result is used to prove that every graph of genus g hasa degenerate coloring with O(g3/5) colors.

Simon SpacapanUniversity of Maribor, SloveniaSimon Fraser [email protected]

Bojan MoharUniversity of [email protected]

MS1

Path Spectrum Sets

A path in a graph is maximal if it is not a proper subpath ofany other path of the graph. The Path Spectrum of a graphis the set of lengths of all maximal paths in the graph. A setof positive integers is an Absolute Path Spectrum if thereare an infinite number of graphs with that Path Spectrum.Known results on the path spectrum will be summarized,some new results on Non-Path Spectrum Sets, and new

38 DM08 Abstracts

results on Absolute Path Spectrum Sets will be presented.

Ralph FaudreeUniversity of MemphisDepartment of Math [email protected]

MS1

F -Avoiding Hamiltonian Graphs

Let G be a graph and H be a subgraph of G. If G containsa hamiltonian cycle C such that E(C) ∩ E(H) is empty,we say that C is an H-avoiding hamiltonian cycle. Let Fbe any graph. If G contains an H-avoiding hamiltoniancycle for every subgraph H of G such that H ∼= F , thenwe say that G is F -avoiding hamiltonian. In this talk,we give minimum degree and degree-sum conditions whichassure that a graph G is F -avoiding hamiltonian for variouschoices of F . If time permits, we will also undertake a briefdiscussion of F -avoiding pancyclic graphs.

Michael FerraraDept. of Theoretical and Applied MathematicsUniversity of [email protected]

MS1

Strong Connectivity and Cycle Structure

In this talk we consider several new developments on strongconnectivity questions in graphs. For a fixed multigraphH, possibly containing loops, with V (H) = {h1, . . . , hk}we say a graph G is H-linked if for every choice of k ver-tices v1, . . . , vk in G, there exists a subdivision of H in Gsuch that vi represents hi (for all i). Using a k matching asthe graph H, we obtain the well-known idea of a k-linkedgraph. An H-immersion in G is similar to H-linkage, ex-cept that the paths in G, playing the role of the edges ofH, are only required to be edge disjoint. We determineminimum degree conditions for a graph G to contain anH-linkage. We further generalize these results to find con-ditions for a graph G to contain an H-immersion with abounded number of vertex repetitions on any choice of kvertices. We then show applications of this work to cyclequestions. This talk spans work in several papers and withseveral sets of coauthors.

Ronald GouldDept. of Mathematics and Computer ScienceEmory [email protected]

MS1

Cycle Lengths Occurring in Hamiltonian Graphs

A simple graph of order n is said to be pancyclic if it con-tains cycles of all lengths fro 3 to n. We consider hamilto-nian graphs with two vertices having ”large” degree sum.The degree sum that insures that the graph is pancyclicwill be given. We also consider the problem of determiningwhat cycle lengths must be present in the graph when thisdegree sum condition is reduced. This work generalizes

results of Schmeichel and Hakimi.

Michael JacobsonDept. of MathematicsUniversity of Colorado at [email protected]

MS1

Placing Vertices and Subgraphs on Cycles of Pre-scribed Lengths

Given a graph G of sufficiently large order, we provide avariety of sharp degree conditions for the existence of cy-cles with additional properties. In particular, one of ourtheorems provides sharp degree sum conditions for the ex-istence of vertex disjoint cycles, each of which has a pre-scribed length and contains a prescribed vertex. This, andsimilar results, will be presented along with current andfuture work in this area.

Colton MagnantEmory [email protected]

MS2

The First Three Levels of an Order PreservingHamiltonian Path in the Subset Lattice

A path {S1, . . . , Sk} in the lattice of subsets of {1, . . . , n}is monotone, if S1 = ∅ and for every i, either (a) everysubset of Si appears among the sets S1, . . . , Si−1, or (b)only one (say S) does not, furthermore Si+1 = S. It is notknown whether there is a monotone Hamiltonian Path inthe subset lattice. We show that there exist a monotonepath that contains all sets of size at most 3 by providingan explicit construction.

Csaba Biro, David HowardGeorgia Institute of [email protected], [email protected]

MS2

Maximal Antichains in Finite Partially OrderedSets

An antichain in a (partially) ordered set X is an unorderedsubset, and a fibre of X intersects every maximal antichain.There are well-known combinatorial problems involvingenumeration of antichains, such as Dedekind’s problem.More recent questions involve enumeration of maximal an-tichains and estimates of minimum fibre sizes. We willpresent results and problems of these types, along withsome natural graph and hypergraph analogues.

Dwight DuffusEmory UniversityMathematics and Computer Science [email protected]

MS2

Large Families of Subsets Avoiding a Given Con-

DM08 Abstracts 39

figuration

Translating Turan-type questions to ordered sets, we areinterested in the maximum size La(n,H) of a family of sub-sets of the set {1, 2, . . . , n}, subject to the condition thata certain configuration (subposet H) is excluded. For in-stance, Sperner’s Theorem solves the problem for H beinga two-element chain. We survey results of this kind, includ-ing bounds when H is the four-element N poset (joint withGyula O.H. Katona) or a more general height two poset.

Jerrold R. Griggs, Linyuan Lincoln YuUniversity of South [email protected], [email protected]

MS2

Characterizing Posets with Linear Discrepancy 2

In a linear extension of a poset there exists a maximumdistance between any two incomparable elements in theextension. The discrepancy of a poset is defined to be theminimum of these maximums over every linear extensionfor a poset. A k-discrepancy-irreducible poset is defined tobe a poset such that if any element is removed from theposet there exists a linear extension such that the distancebetween any two incomparable elements is no more thank − 1. In this paper we show the family of 3-discrepancy-irreducible posets and classify all posets of discrepancy 2.

David Howard, Mitch Keller, Stephen YoungGeorgia Institute of [email protected], [email protected],[email protected]

MS2

Duality for Semiantichains and Unichain Coveringsin Products of Special Posets

A semiantichain in a product of posets is a family whosemembers are comparable only if they differ in both coor-dinates. A unichain is a chain that is constant in one co-ordinate. Saks and West conjectured that for every prod-uct of partial orders, the maximum size of a semiantichainequals the minimum number of unichains needed to coverthe product. We prove this when both factors have width2. We also use the characterization of product graphs thatare perfect to prove it when both factors have height 2.Finally, we make some observations about the case whereboth factors have dimension 2.

Qi LiuWharton SchoolUniversity of [email protected]


MS3

Eigenvalues of 2-Edge Coverings

A 2-edge-covering between G and H is an onto homomor-phism from the vertices of G to the vertices of H so thateach edge is covered twice and edges in H can be lifted backto edges in G. We show how to compute the spectrumof G by computing the spectrum of two smaller graphs,namely a (modified) form of the covered graph H and an-other graph which we term the anti-cover. This is done forboth the adjacency matrix and the normalized Laplacian.We also give an example of two anti-cover graphs whichhave the same normalized Laplacian, and give some simpleapplications.

Steve ButlerUC San [email protected]

MS3

Matchings and Connectivity From Eigenvalues

I will present some old and new connections between theeigenvalues of a graph and its matching number and con-nectivity.

Sebastian CioabaUC San [email protected]

MS3

The Integral Trees of Index 3

A tree is called integral if all its adjacency eigenvalues areintegers. The main results states that there are exactlyeleven integral trees with spectral radius (= index) 3. Thisresult is established by a complete computer search. In thetalk we will present the theoretic tools that were used tomake the search space small enough.

Willem HaemersTilburg UniversityThe [email protected]

MS3

Using Eigenvalues of Graphs to Solve Problems inDesign Theory

In this talk I will give a number of examples of a problemin design theory that can be rephrased as a question abouta graph. For all of these examples bounds on the size ofa design can be found from an eigenvalue bound from theappropriate graph. The problems I am particularly inter-ested in are related to the Erdos-Ko-Rado theorem, thistheorem gives an upper bound on the size of an intersect-ing set system and describes exactly which systems meetthis bound. There are a surprising number of extensions ofthis famous theorem where the bound can be found usingeigenvalue bounds on an appropriate graph.

Karen MeagherUniversity of Regina

40 DM08 Abstracts

Saskatchewan, [email protected]

MS4

Even Cycles in Graphs

A seminal result of Whitney characterizes when two graphshave the same set of cycles (where cycles are viewed as setsof edges). Namely, two graphs have the same cycle if andonly if one can be obtained from the other by repeatedlyrearranging the graph along one- and two-vertex cutsets.We are interested in characterizing when two graphs havethe same set of even cycles (a cycle is even if it contains aneven number of edges). We can give a Whitney type theo-rem for the case where the graphs have high connectivityor for the case where they contain a sufficient number ofpairwise disjoint odd cycles. This is joint work with IrenePivotto and Paul Wollan.

Bertrand Guenin, Irene PivottoUniversity of [email protected], [email protected]

Paul WollanUniversity of [email protected]

MS4

An Algorithm for Testing Group-Labeled Minors

We discuss the k-linkage problem in group-labelled graphs.At the time of writing this abstract, we believe that we havean efficient algorithm for solving the problem for any fixedfinite abelian group, although we have not rigorously veri-fied all of the details. This is joint work with Jim Geelen,University of Waterloo.

Jim Geelen, Tony HuynhUniversity of [email protected], [email protected]

MS4

Duality of Width Parameters

Adapting the method introduced in Graph Minors X, wepropose a new proof of the duality between the bramble-number of a graph and its tree-width. The technique sim-plifies the proof of bramble/tree-width duality since it doesnot rely on Menger’s theorem. One can also derive from itall known dual notions of other classical width-parameters.Finally, it provides a dual for matroid tree-width.

Omid AminiMax Planck Institut fur [email protected]

Frederic [email protected]

Nicolas NisseUniversidad de [email protected]

Stephan ThomasseUniversity of Montpellier 2 - [email protected]

MS4

Obstructions For Rank-Width Two

We discuss obstructions for graphs of rank-width 2 in termsof pivot-minors. Previously it was known that those ob-structions can have at most (63 − 1)/5 = 43 vertices. Weprove that 43 can be improved to 16. The proof actually de-velops a chain theorem for 4-prime graphs. 4-prime graphsare prime graphs in which every cut of cut-rank ≤ 2 hasone side of at most 2 vertices. This is analogous to Hall’sresult on a chain theorem for 4-connected matroids.

Sang-Il [email protected]

MS5

Delaunay Refinement for Manifold Approximation

Delaunay refinement is a greedy technique for constructingprovably good approximations of manifolds of small dimen-sions. The talk will cover some recent results in surface andvolume mesh generation, anisotropic mesh generation andmanifold reconstruction. The algorithms rely on the con-cept of Delaunay triangulation restricted to a manifold andon the related concept of the witness complex introducedby de Silva.

Jean-Daniel BoissonnatINRIA, [email protected]

MS5

The Geometric Centre of Unit Disc Graphs

Motivated by the gateway placement problem in wirelessnetworks, we consider the geometric k-centre problem onunit disc graphs: given a set of points P in the plane, find aset F of k points in the plane that minimizes the maximumgraph distance from any vertex in P to the nearest vertexin F in the unit disc graph of P ∪ F . We describe efficientpolynomial-time solutions to this problem for any fixed k,and consider generalizations on related intersection graphs.

Stephane Durocher, Krishnam Raju JampaniUniversity of [email protected], [email protected]

Anna LubiwUniversity of Waterloo, [email protected]

Lata NarayananConcordia [email protected]

DM08 Abstracts 41

MS5

Simultaneous Embedding of Planar Graphs

Traditional problems in graph visualization deal with a sin-gle graph while simultaneous graph visualization involvesmultiple related graphs. In the latter case nodes are placedin the same locations in all graphs and the graphs are si-multaneously embeddable if crossing-free drawings for eachgraph can be found. We present polynomial time algo-rithms for simultaneous embedding of several classes of pla-nar graphs and prove that some classes of graphs cannotbe simultaneously embedded.

Stephen KobourovDepartment of Computer ScienceUniversity of [email protected]

MS5

Multi-Agent Planning and Crowd Simulation: Re-cent Advances and Challenges

I’ll briefly present the geometric algorithms we have devel-oped for real-time multi-agent planning and collision avoid-ance and crowd simulations, then conclude with discussionson open research challenges for geometric computing inthese areas.

Ming C. LinUniversity of North [email protected] her

MS5

Flow-Based Methods in Manifold Reconstruction

We introduce and survey recent developments in the so-called ”flow-based” techniques in manifold reconstructionand related problems. These techniques use the continuousflow map that results from the integration of a generalizedgradient of the distance function induced by a sample ofa submanifold of a Euclidean space as a tool for design-ing manifold reconstruction algorithms that can guaranteetopological type of their output.

Bardia SadriComputer Science Dept.Duke [email protected]

MS6

Critical Non-Colorings of the 600-Cell ProvingBell’s Theorem

In recent years several sets of rays have been discoveredin four and more dimensions that provide non-coloringproofs of the Bell-Kochen-Specker (BKS) and Bell nonlo-cality theorems. This talk will give an introduction to thisarea of research, casting it as a problem in combinatorialgeometry. I will then introduce a set of 60 rays derivedfrom the 600-cell and show how they can be used to givenon-coloring proofs of the Bell theorems. The 600-cell isa four-dimensional regular polytope with 120 vertices dis-tributed symmetrically on the surface of a four-dimensional

sphere; the vertices come in antipodal pairs, and the 60 dis-tinct directions from the center to the vertices yield the 60rays used in the Bell proofs. My non-coloring proofs makeessential use of the Reyes configurations embedded withinthe 60 rays (a Reyes configuration is a set of 12 points and16 lines with the property that four lines pass through anypoint and three points lie on any line). By a critical non-coloring I mean one that ceases to be viable if even a singleray is deleted from it. The talk will focus mainly on themathematics of the non-coloring proofs, but some possibleapplications will also be mentioned.

Padmanabhan AravindDepartment of PhysicsWorcester Polytechnic [email protected]

MS6

Title Not Available At Time of Publication

Abstract not available at time of publication.

Alexei AshikhminFundamental Mathematics GroupAlcatel-Lucent Bell [email protected]

MS6

Mutually Unbiased Bases and Association Schemes

In this talk, we explore the connection between real mu-tually unbiased bases and association schemes, focusing onthe Q-polynomial case.

William J. MartinDepartment of Mathematical SciencesWorcester Polytechnic [email protected]

MS6

On Quantum Algorithms for the Hidden SubgroupProblem

Most exponential speedups in quantum computing are ob-tained by solving instances of the so-called hidden sub-group problem (HSP). A common feature of the resultingquantum algorithms is to make measurements of quantummechanical states that encode a secret subgroup of knowngroup. While factoring of integers and the computationof discrete logarithms in abelian groups can be solved ef-ficiently within this framework on a quantum computer,a tantalizing open problem is whether it also lends itselfto an efficient quantum algorithm for the graph isomor-phism problem. In this talk we address this question byanalyzing the approach to graph isomorphism via HSPs inthe symmetric group of all permutations. We show thathighly entangled measurements on at least Omega(n logn) quantum states are necessary to get useful information,matching an information theoretic upper bound. This isjoint work with Sean Hallgren and Pranab Sen.

Martin RoettelerQuantum IT Group

42 DM08 Abstracts

NEC Laboratories [email protected]

MS6

Quantum State Tomography and 2-Designs

Optimal measurements for quantum state tomography arecharacterized by 2-designs. This result is true for sev-eral different types of measurements: mutually unbiasedbases (Wootters and Fields, 1989; Klappenecker and Roet-teler, 2005), general rank-one measurements (Scott, 2006),orthogonal rank-one measurements, and “two-outcome”measurements. In this talk, we explain this connectionand describe some new constructions. This talk is basedon joint work with Andrew Scott, Martin Roetteler, andChris Godsil.

Aidan RoyQuantum Information ScienceUniversity of [email protected]

MS7

On Graph Classes and Powers of Graphs

Chordal graphs are among the most important graphclasses. They are well known for nice structural proper-ties and various applications. They have many facets suchas tree structure, clique separators, perfect elimination or-derings, Helly property, convexity, and various other as-pects. It is well known that odd powers of chordal graphsare chordal. In this talk we discuss similar properties ofvarious other classes which are closely related to chordalgraphs, and we also consider natural variants of powers ofgraphs.

Andreas BrandstaedtUniversitat [email protected]

MS7

A Characterization of b-Perfect Graphs

A b-coloring is a coloring of the vertices of a graph suchthat each color class contains a vertex that has a neigh-bor in all other color classes. The b-chromatic number ofa graph G is the largest integer k such that G admits ab-coloring with k colors. A graph is b-perfect if the b-chromatic number is equal to the chromatic number forevery induced subgraph H of G. A graph is minimally b-imperfect if it is not b-perfect and every proper inducedsubgraph is b-perfect. We conjecture that a graph is b-perfect if and only if it does not contain a graph from acertain list of 22 graphes as an induced subgraph. Wepresent several partial results on this conjecture. This isjoint work with Frederic Maffray and Meriem Mechebbek.

Chinh HoangWilfrid Laurier UniversityDept of Physics and Computing [email protected]

MS7

On Probe Graph Classes


Ton KloksSchool of Computing, Faculty of EngineeringUniversity of Leeds, [email protected]

MS7

On Graph Classes Determined by Convexity inGraphs

A collection M of subsets of a finite set V is an alignmentof V if M is closed under intersections and contains bothV and the empty set; in that case the elements of M areconvex sets. The convex hull of S ⊆ V is the smallestconvex set that contains S. For X ∈ M a point x ∈ X isan extreme point for X if X \{x} ∈ M. An alignment withthe additional property that every convex set is the convexhull of its extreme points is a convex geometry. The Steinerinterval of a set S of vertices in a connected graph G is thecollection of all vertices that belongs to some Steiner treefor S, i.e., a sub-tree of G of smallest size containing S. Wecharacterize graph classes that form the convex geometriesof alignments that are defined in terms of Steiner intervals.

Ortrud R. OellermannThe University of [email protected]

Morten H. NielsenUniversity of [email protected]

MS7

On Helly Classes of Graphs

In 1923, Eduard Helly published a celebrated theorem,which originated the well known Helly property. Say thata family of subsets has the Helly property when every sub-family of it, formed by pairwise intersecting subsets, con-tains a common element. There are many generalizationsof this property which are relevant to parts of mathematicsand computer science, and which motivated the definitionof some different classes of graphs and hypergraphs. In thistalk, we survey computational aspects of the Helly prop-erty. In the context of discussing the recognition of theseclasses of graphs and hypergraphs, we describe algorithmsfor solving different problems arising from the basic Hellyproperty, or else present NP-hardness results.

Mitre DouradoUniversidade Federal do Rio de [email protected]

Fabio ProttiInstituto de Matematica e NCEUniversidade Federal do Rio de [email protected]

Jayme L. SzwarcfiterUniversidade Federal do Rio de Janeiro

DM08 Abstracts 43

[email protected]

MS8

Biplanar Crossing Numbers

The crossing number cr(G) of a graph G is the minimumnumber of edge crossings with which the graph can bedrawn in the plane. The biplanar crossing number of agraph G is the minimum of cr(G1)+cr(G2), where G1 andG2 are graphs on the same vertex set as G such that theunion of their edge set is the edge set of G. We examinehow much the crossing number and the biplanar crossingnumber of a graph can differ. This is joint work with LaszloSzekely, Ondrej Sykora and Imrich Vrto.

Eva CzabarkaDepartment of MathematicsUniversity of South [email protected]

MS8

Hanani-Tutte on the Projective Plane

Hanani (1934), Tutte (1970), and others have given proofsfor the following theorem: In any planar drawing of a non-planar graph, there are two non-adjacent edges that crossan odd number of times. We prove the analogous resultfor the projective plane: For any drawing on the projectiveplane of a graph that cannot be embedded on the projec-tive plane, there are two non-adjacent edges that cross anodd number of times.

Michael J. PelsmajerIllinois Institute of TechnologyDepartment of Applied [email protected]

Despina StasiMathematics, Statistics, and Computer ScienceUniversity of Ilinois at [email protected]

Marcus SchaeferDePaul UniversitySchool of [email protected]

MS8

On the Induced Matching Problem

We study extremal questions on induced matchings in sev-eral natural graph classes. We argue that these questionsshould be asked for twinless graphs, that is graphs notcontaining two vertices with the same neighborhood. Weshow that planar twinless graphs always contain an inducedmatching of size at least n/40 while there are planar twin-less graphs that do not contain an induced matching ofsize (n + 10)/27. We derive similar results for outerpla-nar graphs and graphs of bounded genus. These extremalresults can be applied to the area of parameterized compu-tation. For example, we show that we can decide in timeO(91k + n) whether a planar graph contains an induced

matching of size at least k.

Iyad KanjDePaul [email protected]

Michael J. PelsmajerIllinois Institute of [email protected]

Marcus SchaeferDePaul UniversitySchool of [email protected]

MS8

Open Problems on Crossing Number

The talk will review open problems about crossing num-bers, starting with the 50 years old Zarankiewicz Conjec-ture, and arriving at more recent problems.

Laszlo SzekelyUniversity of South [email protected]

MS9

Submodular Approximation: Sampling-based Al-gorithms and Lower Bounds

We introduce several natural optimization problems us-ing submodular functions and give asymptotically tight orclose upper and lower bounds on approximation guaran-tees achievable using polynomial number of queries to afunction-value oracle. The optimization problems we con-sider are submodular load balancing, submodular spars-est cut, submodular balanced cut, submodular knapsack.We also give a new lower bound for approximately learn-ing a monotone submodular function; and show that muchtighter lower bounds will require a different approach.

Lisa FleischerDartmouth [email protected]

MS9

Network Decompositions with Applications toAnomynity

Some anonymity protocols for communication over the in-ternet rely on suitable decompositions of underlying graphsand the several objective functions associated with theseprotocols relate to well-known graph parameters. We dis-cuss these relations for an anonymity protocol based oncycle-coverings and present some related results.

Dieter RautenbachInstitut fuer MathematikTU [email protected]

44 DM08 Abstracts

MS9

Robust Optimization of Networks

We consider several models of how to design a cheapestnetwork that supports any traffic specified by a demandmatrix [dij ] in a given set, or polytope. We show thatthe complexity of this problem depends on the choice ofpolytope, as well as whether we are allowed to adapt ourroutings in response to changing traffic patterns. We alsoshow some empirical results that show the applicability ofthese ideas to data networks.

Bruce ShepherdMcGill [email protected]

MS9

Steiner Trees in Chip Design

Finding shortest Steiner trees in grids and packing Steinertrees subject to various constraints are two major subprob-lems in chip design. The networks in which we look forSteiner trees are huge, and instances need to be solved fast.We will outline the role of Steiner trees in this applicationand present new algorithms with surprising performance.

Jens VygenUniversity of [email protected]

MS10

Graph and Hypergraph Models and Algorithms forSparse Matrix-Vector Multiplication

Combinatorial models using graphs and hypergraphs arefrequently used in scientific computing. We give anoverview of models and algorithms for an important kernel,parallel sparse matrix-vector multiplication. This is a sur-prisingly rich model problem, and relevant for other com-putations as well. We present recent research in 2D datadistribution methods, using both graphs and hypergraphs.We demonstrate these techniques can significantly reducecommunication volume compared to standard methods.

Erik G. BomanSandia National Labs, NMScalable Algorithms [email protected]

Michael M. WolfUniversity of Illinois, [email protected]

MS10

Approximation Algorithms for Vertex WeightedMatching

We describe a 2/3-approximation algorithm for computingmaximum vertex-weighted matchings in bipartite graphs.This approximation ratio is better than what has beenachieved for the edge-weighted matching problem. Ourwork leads to new algorithms for computing optimalvertex-weighted matchings. The study of vertex-weighted

matchings also unveils the rich combinatorial structure ofthe problem.

Alex PothenPurdue UniversityDept of Computer [email protected]

Florin DobrianColumbia [email protected]

Mahantesh HalappanavarOld Dominion [email protected]

MS10

Graph Sparsification by Effective Resistances

Sparsification is the task of approximating a given graphby a sparse graph. We consider two graphs to be similar ifthe spectra of their Laplacian matrices are close; this no-tion of approximation preserves many combinatorial prop-erties, such as the weights of all cuts in a graph, and alsohas applications in solving certain systems of linear equa-tions quickly. We show that every graph on n vertices canbe approximated by a subgraph with O(nlogn) edges andconjecture the existence of sparsifiers with O(n) edges, not-ing that these would generalize expander graphs, which areconstant degree sparsifiers for the complete graph.

Nikhil Srivastava, Daniel SpielmanYale [email protected], [email protected]

MS10

Open Problems in Combinatorial Preconditioning

The talk will describe important open problems in combi-natorial preconditioning. In particular, I will describe theproblem of constructing combinatorial preconditioners formulti-commodity flow problems. I will describe the rela-tionship of matroid bases to preconditioners, the particularmatroid that arises in multi-commodity flow problems, anddifficulty of using maximum-weight matroid bases as pre-conditioners in this case.

Sivan A. ToledoMIT (on sabbatical from Tel Aviv University)[email protected]

MS11

The Edge Crossing Graph

Given a geometric graph G, the edge crossing graph of G,denoted by EX = EX(G), captures the relation of edgescrossing. Specifically, EX is the graph defined by V (EX) =E(G) and E(EX) = {ef ∈ E(G) : e and f cross in G}. Thegoal is to relate properties of EX(G) with χ(G). It is openwhether given an abstract graph H there exists G suchthat EX(G) ∼= H. This talk reports partial results on both

DM08 Abstracts 45

these issues.


MS11

Planar Decompositions and the Minor CrossingNumber of Graphs

The minor crossing number of a graph G is the minimumcrossing number of a graph that contains G as a minor.Using planar graph decompositions, we prove that for everygraph H there is a constant c, such that every graph G withno H-minor has minor crossing number at most c|V (G)|.This result complements an earlier result by Wood andTelle about the ordinary crossing number, which in thesame context equals at most cΔ(G)2|V (G)|, where Δ(G)denotes the maximum degree of G.

Drago BokalFaculty of natural science and mathematicsUniversity of Maribor, [email protected]

Gasper FijavzFaculty of computer and information scienceUniversity of [email protected]

David R. WoodDepartment of applied mathematicsPolytechnic University of [email protected]

MS11

The Rectilinear Crossing Number of the CompleteGraph:Closing in (or are We?)

The problem of determining the rectilinear crossing num-ber of the complete graphs Kn is an open classical problemin discrete geometry. A major breakthrough was achievedin 2003 by two teams of researchers working independently(Abrego and Fernandez-Merchant; and Lovasz, Veszter-gombi, Wagner and Welzl), revealing and exploiting theclose ties of this problem to other classical problems, suchas the number of convex quadrilaterals in a point set, thenumber of (≤ k)–sets in a point set, the number of halvinglines, and Sylvester’s Four Point Problem. Since then, wehave seen a sequence of improvements both from the lowerbound and from the upper bound sides of the problem, andnowadays the gap between these bounds is very small. Ouraim in this talk is to review the state of the art of theseproblems.

Gelasio SalazarInstituto de FisicaUniversidad Autonoma de San Luis [email protected]

MS12

PTAS for Max. Independent Set on Planar Graphs

through Sherali-Adams System

In the eighties a few papers described (different) meth-ods of getting a polytime approximation scheme (PTAS)for certain NP-hard graph problems, most notably vertex-cover and maximum independent set, when the input isplanar. All these methods are combinatorial. In this workwe show how such PTAS emerge simply from (i) startingwith the integer programming for the problems (ii) relax-ing them and applying the Sharali-Adams lift and projectsystem. The integrality gaps such relaxations achieve are1+1/k when the running time is exp(k). Comparing themto the combinatorial algorithms mentioned above, our al-gorithms have the benefit of robustness. Specifically, evenif the graph is not quite planar but close to being one, theintegrality gaps are still converging to 1 as above, whereasthe combinatorial algorithms don’t seem to handle this casewell.

Avner Magen, Mohammad MoharramiUniversity of [email protected], [email protected]

MS12

Lift-and-Project Operators for Set Packing and SetCovering Problems on Circulant Matrices

When working with the set packing problem on circulantmatrices, widely studied polyhedra appear: the stable setpolytope of webs and antiwebs and their clique relaxations.Also, circulant matrices and their blockers have been fairlystudied on the set covering problem because they are sup-posed to be closely connected with mni matrices. In thiswork, symmetries and asymmetries between set packingand covering problems on circulant matrices are studiedthrough the performance of BCC and N+ operators.

Graciela NasiniUNR, ArgentinaDept of [email protected]

MS12

Exponential Lower Bounds and Integrality Gapsfor Tree-Like Lovasz-Schrijver Procedures


Toniann PitassiUniversity of [email protected]

MS12

Lift-and-Project Methods and the Approximabilityof Vertex Cover

Lift-and-project methods are used to iteratively tightenlinear and semidefinite relaxations by ”lifting” the relax-ation to a higher dimensional space, adding valid con-straints and then projecting back down to the originalspace. Several recent works study the potential of suchmethods for designing approximation algorithms. Motivat-ing this work was the realization that the relaxations un-

46 DM08 Abstracts

derlying such breakthrough approximation algorithms likethose for Max-Cut (Goemans-Williamson) and Sparsest-Cut (Arora-Rao-Vazirani) are all efficiently derivable usinglift-and-project methods. One optimization problem thathas attracted attention from approximation algorithms re-searchers working with lift-and-project methods is the clas-sic Vertex Cover problem. While there exist simple algo-rithms approximating Vertex Cover within a factor of 2,concerted efforts by many researchers have failed to yieldbetter approximations. The consensus was that semidefi-nite programming and, in particular, lift-and-project meth-ods were our ”last best hope” for obtaining a better ap-proximation for Vertex Cover. We deal a blow againstthis hope: We show that no polynomial-time semidefiniteprogramming relaxations obtained using the powerful LS+lift-and-project method of Lovasz and Schrijver can yieldbetter than a 2-o(1) approximation for Vertex Cover.

Iannis Tourlakis, Konstantinos Georgiou, Avner Magen,Toniann PitassiUniversity of [email protected], [email protected],[email protected], [email protected]

MS13

Progress Using the Substitution Method to BoundCritical Probabilities

One of the central problems of percolation theory is deter-mination of critical probability values, pc, for large classesof percolation models. This is a difficult, and largely un-solved, problem. In this talk we discuss recent progressagainst this problem using the substitution method. Wehave adopted new computational methods to make thesubstitution method tractable over larger regions. Thesemethods include exploitation of symmetry, using networkflow algorithms to prove stochastic ordering, and exploita-tion of non-crossing partitions for certain classes of prob-lems. Using the new computations we greatly improvedbounds on pc for a number of models, but fell short ofachieving our challenge goal: disproving Tsallis’s conjec-tured value of pc for the (3, 122) lattice.

William MayApplied Signal [email protected]

MS13

Approximation Formulas for Percolation Thresh-olds

The talk will discuss approximation formulas for bond andsite percolation thresholds, criteria for evaluating them,and techniques for improving them. Inspired by the bond-to-site transformation, a formula for the average degree ofa line graph in terms of the degrees in the original graph isderived and used to improve percolation threshold approx-imation formulas for two-dimensional lattices.

John WiermanDept of Applied Mathematics and StatisticsJohns Hopkins [email protected]

MS13

An Introduction to Percolation Thresholds

In preparation for the later talks in the mini-symposium,this survey talk will introduce site and bond percolationmodels and the concept of the critical probability or perco-lation threshold. Classes of periodic graphs, relationshipsbetween them, and previous exact values, estimates, andbounds for their percolation thresholds will be discussed.

John WiermanDept of Applied Mathematics and StatisticsJohns Hopkins [email protected]

MS13

The Triangle-Triangle Transformation and ExactPercolation Thresholds


Robert ZiffDept of Chemical EngineeringUniversity of [email protected]

MS14

Minimum Span Graph Labellings with SeparationConditions

The theory of integer vertex λ-labellings of a graph modelsthe efficient assignment of channels to a network of trans-mitters. One seeks the minimum span of a labelling suchthat labels for vertices at distance i are separated by atleast a specified amount ki. We present an overview of theconsiderable recent progress and future challenges, both onthe original case (k1 = 2 and k2 = 1) and on more generalmodels of real-number labellings of infinite graphs.

Jerrold R. GriggsUniv. of S. Carolina, [email protected]

MS14

L(p, q)-Labelling of Graphs

An L(p, q)-labelling of G is an integer assignment f tothe vertex set V (G) such that: |f(u) − f(v)| ≥ p, ifdist(u, v) = 1, and |f(u) − f(v)| ≥ q, if dist(u, v) = 2.λp,q(G) is the least integer k such that g has an L(p, q)-labelling in {1, . . . k}. In 1992, Griggs and Yeh conjec-tured that λ2,1(G) ≤ Δ2 + 1, where Δ is the maxi-mum vertex degree in G. We prove this conjecture forsufficiently large Δ. In 1977, Wegner conjectured thatλ1,1(G) ≤

⌊32

Δ⌋

+ 1 for a planar graph G if Δ ≥ 8. We

show that λ1,1(G) ≤ (1 + o(1)) 32

Δ. These two resultsgeneralise to L(p, q)-labelling and list-colouring.

Frederic HavetProjet Mascotte, I3S(CNRS/UNSA) and [email protected]

Jan Van Den Heuvel

DM08 Abstracts 47

Department of MathematicsLondon School of [email protected]

Colin McDiarmidUniversity of Oxford, [email protected]

Bruce ReedMcGill [email protected]

Jean-Sebastien SereniIntitute for nTheoretical Computer ScienceCharles [email protected]

MS14

Ramsey and Survival Games

The (c, s, t)-Ramsey game G(�, ∫ ,�) is played by Builderand Painter on a set V of vertices determined by Builder.Starting from the empty graph, Builder constructs an s-uniform hypergraph H by adding edges one by one. Whenan edge is added, Painter gives it one of c colors. Builderwins if Painter creates a monochromatic copy of Kt

s, thecomplete s-uniform hypergraph on t vertices. By Ramsey’sTheorem, Builder can win by building a large completehypergraph (with huge chromatic number). We prove forall c, s, t that Builder can win G(�, ∫ ,�) when required tomaintain χ(H) ≤ χ(Kt

s). More strongly, we give a natu-ral definition of coloring number for hypergraphs and showthat chromatic number can be replaced by coloring num-ber in this theorem. The proof is based on analysis of the(p, s, t)-survival game S(√, ∫ ,�) played by Presenter and

Chooser on a set V of vertices determined by Presenter.The players build an s-uniform hypergraph H. In the ithround Presenter plays a p-subset Pi ⊆ V and chooser se-lects an s-subset Xi ⊆ Pi. The vertices in Pi − Xi, andany edges containing them, are removed from H and theedge Xi is added to H. Presenter wins if H ever containsa copy of Kt

s. We prove that Presenter can win S(√, ∫ ,�)

for all c, s, t.

H. A. Kierstead, Goran KonjevodArizona State [email protected], [email protected]

MS14

Relaxed Colorings of Graphs

A coloring of the vertices of a graph G is called C-relaxedif all monochromatic connected components have orderbounded by C. The usual proper coloring is then equiv-alent to 1-relaxed coloring. We investigate Brooks-typetheorems for this relaxed coloring concept and its gener-alizations, mostly for the case when C is a constant. Wediscuss both extremal graph theoretic and computationalcomplexity theoretic aspects.

Robert Berke, Tibor SzaboETH Zurich, Switzerland

[email protected], [email protected]

MS14

On-Line Ramsey Theory for Bounded DegreeGraphs

The on-line Ramsey game is played by Builder and Painter.Builder presents one edge at a time, which Painter mustcolor red or blue. Builder wins if a monochromatic copy ofG appears. The on-line degree Ramsey number of G is theleast k such that Builder can force G without giving degreemore than k to any vertex. The value is 3 for paths (withat least four vertices), at most 2Δ(G)−1 for trees (which issharp), and usually 4 for cycles. We also describe generaltechniques useful for proving upper and lower bounds.

Jane Butterfield, Tracy Grauman, Bill Kinnersley, KevinMilans, Christopher StockerUniversity of [email protected], [email protected],[email protected], [email protected],[email protected]


MS15

Phylogenetic Analysis of Molecular InteractionNetworks

In systems biology, availability of high throughput molecu-lar interaction data provides novel opportunities for model-ing and analyzing cellular organization through graph the-oretical abstractions. Comparative analysis of these in-teractions across diverse species indicate that conserva-tion/divergence of networks can be used to understandfunctional evolution from a systems perspective. In thistalk, we discuss the problem of reconstructing phylogeniesbased on interaction data. For this purpose, we proposea modularity based approach that alleviates intractablegraph comparison problems, allow accounting for noise andmissing data, and provide insights on the evolution of mod-ularity in biological systems. We show on a comprehensivecollection of simulated and real networks that the proposedtechnique is promising in accurately capturing evolutionarydistances between diverse species, is robust to noise, andoutperforms existing phylogenetic network analysis meth-ods.

Mehmet KoyuturkCase Western Reserve UniversityDepertment of Electrical Engineering and [email protected]

MS15

Identification Characters with Minimal Homoplasyfor Inferring Maximum Parsimony Trees

Parsimony methods infer phylogenetic trees by minimizingnumber of character changes required to explain observed

48 DM08 Abstracts

differences. Parsimony methods may however be confusedby occurrence of homoplasy - a situation where speciesshare the same properties (characters) but these charactersare not derived from a common ancestor. Instead, they arearrived to by two independent evolutionary pathways (e.g.wings in birds and bats). In the presence of excess of ho-moplasy, parsimony methods often produce incorrect trees.We discuss graph-theoretical and methods of controllinghomoplasy in the data. Based on these graph-theoreticalinsights, we developed procedure for the identification andanalysis of conserved characters with minimal homoplasyand show the applicability of our approach to solving hardphylogenetic problems. The results discussed in this talkhave been obtained in collaboration with Eugenie Koonin,Igor Rogozin, and Jie Zhang.

Teresa PrzytyckaNCBI, [email protected]

MS15

Analyzing Genome Rearrangements in Evolutionand Cancer

Mutations in genomes range from single letter changes inthe DNA sequence to rearrangements, gains, or losses oflarge pieces of DNA. In particular, mammalian genomescontain extensive segmental duplications, many of whichare complex mosaics of fragments of numerous other seg-mental duplications. A similar phenomenon has recentlybeen observed in cancer genomes that contain complicatedpatterns of overlapping rearrangements and duplications.We describe algorithms to determine a parsimonious se-quence of rearrangement and duplication operations thattransform one genome sequence into another genome se-quence. We use these algorithms to analyze segmental du-plications in the human genome and genome organizationin cancer.

Ben RaphaelBrown [email protected]

MS15

Algorithms for Identifying Potential Targets on Es-sential Pathways and Protein-Protein InteractionNetworks of Pathogens

As pathogens evolve effective schemes to overcome theeffect of antibiotics, the prevalent ”one drug and onedrug target” approach is falling behind. We proposenovel strategies for identifying potential multiple-drug tar-gets in pathogenic PPI networks with the goal of dis-rupting known pathways/complexes. Given a set S ofpathogenic pathways/complexes, we first consider comput-ing the minimum number of proteins (with no human or-thologs) whose removal from the PPI network disrupts allpathways/complexes. Unfortunately even the best approx-imation algorithms for this (NP-hard) problem return toomany targets to be practical. Thus we focus on comput-ing the optimal tradeoff (i.e. maximum ratio) betweenthe number of disrupted essential pathways/complexesand the protein targets. For this ”sparsest cut” prob-

lem, we describe two polynomial time algorithms withrespective approximation factors of |S|andO(sqrt(n))(n :numberofnodes). On the E.coli PPI network with 9 es-sential (signaling) paths from the KEGG database, our al-gorithms show how to disrupt 3 of them by targeting only3 proteins (2 of them essential proteins). We also con-sider the case where there are no available essential path-ways/complexes to guide us. In order to maximize thenumber of disrupted ”potential” pathways/complexes weshow how to compute the smallest set of proteins whoseremoval partitions the PPI network into two almost-equalsized subnetworks so as to maximize the number of poten-tial pathways/complexes disrupted. This approach yields28 potential targets (4 of them known drug targets) onthe E.coli PPI network whose removal partitions it to twosubnetworks with relative sizes of 1 to 5.

Cenk SahinalpSimon Fraser [email protected]

MS15

Function and Topology in Protein Interaction Net-works

In recent years, high-throughput technologies have resultedin large-scale determination of protein-protein interactionsfor several organisms. Computational analyses of the re-sulting interaction networks provide new opportunities forrevealing cellular organization and uncovering protein func-tion and pathways. In this talk, I will discuss some of mygroup’s recent work on the analysis of protein physical in-teraction networks, focusing on the interplay between pro-tein function and network structure.

Mona SinghPrinceton [email protected]

MS16

Small Label Classes in 2-Distinguishing Labelings

A graph G is said to be 2-distinguishable if there is a label-ing of the vertices with two labels so that only the trivialautomorphism preserves the vertex labels. Denote the min-imum size of a label class in such a labeling by ρ(G). Ifwe consider labeling the graph by coloring one label classof vertices red and not coloring the other, ρ tells us theminimum number of vertices we need to color to break allsymmetry. This talk will introduce ρ and give bounds onρ(Qn) and ρ(Kn

3 ). We will see that both are Θ(log n).

Debra L. BoutinHamilton [email protected]

MS16

The Distinguishing Chromatic Number (Part II)

Collins and Trenk introduced the distinguishing chromaticnumber of a graph G, χD(G), as the minimum number ofcolors needed to color the vertices so that (1) the color-ing is a proper graph coloring and (2) the only automor-

DM08 Abstracts 49

phism of the graph which preserves colors is the identity.Thus χD(G) is closely related to both the chromatic num-ber and the distinguishing number of a graph. A naiveapproach to finding an upper bound for the distinguishingchromatic number of a graph would be to expect that itwould be less than the sum of the distinguishing numberplus the chromatic number. In this talk, we provide a tightbound on the distinguishing chromatic number of graphswith abelian automorphism group Γ, where the differencebetween χD(G)−χ(G) can be any nonnegative integer, de-pending on the number of prime power factors of Γ; how-ever, D(G) for graphs with abelian automorphism groupsis always less than or equal to 2.

Karen Collins, Mark HoveyWesleyan [email protected], [email protected]


MS16

Distinguishing Number of Line Graphs of Com-plete Bipartite Graphs

For s ≤ t the distinguishing number of the cartesian prod-uct of complete graphs of sizes s and t (the line graphof a complete bipartite graphs with part sizes s and t) iseither�(t+ 1)1/s� or �(t+ 1)1/s�+ 1. We discuss a proof ofthis result from the perspective of coloring the edges of acomplete bipartite graph and discuss related results.

Garth T. IsaakLehigh UniversityDepartment of [email protected]

MS16

The Distinguishing Chromatic Number (Part I)

Albertson and Collins introduced the distinguishing num-ber of a graph as the minimum number of colors neededto color the vertices so that the only automorphism of thegraph which preserves colors is the identity. The distin-guishing chromatic number, χD(G), is defined similarly, ex-cept that the coloring must also be proper, that is, adjacentvertices must get different colors. In this talk we discussresults about χD(G) including characterizations of χD(G)for various families of graphs, analogues of Brooks’ Theo-rem, and results that relate χD(G) to the automorphismgroup of G.


Karen Collins, Mark HoveyWesleyan [email protected], [email protected]

MS16

Genericity of Distinguishing Number Two

In many different situations, the Motion Lemma impliesthat all but finitely many cases have distinguishing numbertwo, that is having distinguishing number two is generic.Examples are presented from various contexts of groupsacting on sets: automorphism of finite groups, automor-phims of finite vectors spaces, maps, locally finite (infinite)graphs, general transitive actions.

Thomas TuckerColgate [email protected]

MS17

Colorings and Crossings

In a drawing of a graph G, two crossings are dependent ifthey are incident with a common vertex. A set of crossingsis independent if no two are dependent. We conjecture thatif G has a drawing whose crossings are all independent,then G has chromatic number at most 5. We prove thiswhen the crossing number is at most 3. We show that if allcrossings are independent, then the chromatic number is atmost 6. Other connections between crossings and coloringswill be examined.


MS17

List Colouring Hypergraphs

We say that a hypergraph H is L-colourable for a givenassignment {L(v) : v ∈ V (H)} of vertex lists if H has avertex-colouring c such that c(v) ∈ L(v) for each v andno edge of H is monochromatic. The hypergraph H issaid to be k-list-colourable if it is L-colourable for everyL satisfying |L(v)| ≥ k for each v. We consider a num-ber of problems concerning list colouring of hypergraphs,in particular some that are related to certain transversalproblems in hypergraphs.

Penny HaxellUniversity of [email protected]

MS17

Coloring H-Free Hypergraphs

Let H be an r-uniform hypergraph. What is the minimumnumber of edges in an H-free r-uniform hypergraph withchromatic number greater than k? We consider this prob-lem for various H. In the special case when H consists oftwo edges sharing p vertices, we improve the previous bestupper bounds due to Kostochka, Rodl, Mubayi and Tetali.

Tom BohmanCarnegie Mellon UniversityDepartment of [email protected]

50 DM08 Abstracts

Alan FriezeCarnegie Mellon [email protected]

Dhruv MubayiUniversity of [email protected]

MS17

Coloring Problems for Graphs That Arise FromPosets

There are several challenging graph coloring problems thatarise from the study of partially ordered sets. We willpresent updates on three: the maximum chromatic num-ber of a diagram of an interval order of given height; theperformance of first fit coloring on interval graphs; and thechromatic number of arbitrary poset diagrams of boundedheight and size.

W. Tom TrotterGeorgia Institute of [email protected]

MS17

List Colorings of Planar Graphs

A graph G = (V,E) is k-list colorable if for every list as-signment L with |L(v)| ≥ k for all v ∈ V there exists acoloring c of the vertices of G where c(v) ∈ L(v) for allv ∈ V and c(v) �= c(w) for all vw ∈ E(G). It is well-known that every planar graph is 5-list colorable and thereare planar graphs which are not 4-list colorable. Many au-thors tried to find conditions for planar graphs ensuringk-list colorability for k ∈ {3, 4}. The talk gives a recentoverview on results in this field.

Margit VoigtUniversity of Applied [email protected]

MS18

Cycles in Pairwise Balanced Designs with Index 1

The block-intersection graph of a pairwise balanced designis the graph whose vertices are the blocks of the designand whose edges correspond to the pairs of blocks withnon-empty intersection. In this talk, the structure and cy-cle properties of these graphs will be discussed for designswhose minimum block size is 3 and index is 1. The struc-ture of the more general cases that include those designswith minimum block size 2 or index at least 1 will also bediscussed.

Donovan HareThe University of British Columbia, Okanagan, [email protected]

MS18

Near Ucycles Exist for Subsets

A universal cycle (ucycle) for k-subsets of an n-set X is a

cyclic string S of elements from X with the property thatevery k-subset of X appears exactly once as a contiguoussubstring of S. Invented by Chung, Diaconis and Grahamin the early 1990s, few examples are known to exist. Auniversal packing (upacking) replaces the ”exactly once”condition by ”at most once”. A near-ucycle is a upackingfor which the number of k-subsets that do not appear isasymtotically small compared to

(nk

). Here we prove for

all k that near-ucycles exist when n is large enough. Thisis joint work with Curtis, Hines, and Moyer.

Dawn Curtis, Taylor Hines, Glenn Hurlbert, TatianaMoyerArizona State [email protected], [email protected],[email protected], [email protected]

MS18

Hamiltonicity and Restricted Block-IntersectionGraphs of T-Designs

Given a combinatorial design D with block set B, its tra-ditional block-intersection graph GD is the graph havingvertex set B such that two vertices b1 and b2 are adjacentif and only if b1 and b2 have non-empty intersection. Weconsider the S-block-intersection graph, in which two ver-tices b1 and b2 are adjacent if and only if |b1 ∩ b2| ∈ S.As our main result we prove that {1, 2, . . . , t − 1}-block-intersection graphs of t-designs with parameters (v, t+1, λ)are Hamiltonian whenever t ≥ 3 and v ≥ t+3, except pos-sibly when (v, t) ∈ {(8, 5), (7, 4), (7, 3), (6, 3)}.David PikeMemorial University of [email protected]

Robert Vandell, Matthew WalshIndiana-Purdue University, Fort [email protected], [email protected]

MS18

Previous and Current Results on Ordering theBlocks of Designs

In 1989, Ron Graham asked if 1-block intersection graphsof Steiner triple systems are Hamiltonian. This is one ofthe first instances in combinatorics of asking for a particu-lar ordering of the blocks of a design. This Hamiltonicityquestion has been investigated for BIBDs and PBDs withmany parameters. Other research on ordering the blocks ofdesigns include Gray codes and universal cycles of completehypergraphs, configuration orderings and pair designs. Wewill introduce the various orderings that have been re-searched and summarize the results of scientists unable tospeak at this symposium. Additionally we will present newresults from Megan Dewar’s Ph.D. Thesis: rank 3 univer-sal cycles and Gray codes of twofold triples systems, rank2 universal cycles of cyclic BIBDs, and new configurationorderings including huts and partial parallel classes.

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

DM08 Abstracts 51

Megan E. DewarDepartment of National [email protected]

MS18

Single Change Designs

A single change design is a block design whose blocks areordered so that each block can be derived from the preced-ing one by deleting one member and introducing one newmember. We shall discuss single change covering designsand single change BIBDs. There are a number of unsolvedproblems concerning these classes of designs.

Walter WallisSouthern Illinois University at [email protected]

MS19

Optimal Distribution Networks

We describe a simple geometric argument to explain howmaterial-delivering networks are optimally structured. Weoutline implications for energy usage in organisms, thestructure of plants, transportation systems, and river net-works.

Peter DoddsUniversity of [email protected]

MS19

Effects of Community Structure on RespondentDriven Sampling

Abstract: Respondent driven sampling (RDS) is a recentlyintroduced, and now widely used, technique for estimat-ing disease prevalence in hidden populations. The sampleis collected through a form of snowball sampling wherecurrent sample members recruit future sample members.We present respondent-driven sampling as Markov chainMonte Carlo (MCMC) importance sampling, and exam-ine the effects of community structure and recruitmentmethodology on the variance of RDS estimates. Past workon RDS has assumed that the variance of RDS estimatesis primarily affected by segregation between healthy andinfected individuals. We examine an illustrative model toshow that this network feature, while important, in iso-lation tends to significantly underestimate the effects ofcommunity structure on RDS estimates. We also showthat variance is increased by a sample design feature whichallows sample members to recruit multiple future samplemembers. Our observations are further substantiated bynetwork data collected as part of the National Longitudi-nal Study of Adolescent Health.

Sharad GoelYahoo! [email protected]

Matthew SalganikPrinceton University

[email protected]

MS19

The Structure and Dynamics of Social Communi-cation Networks

Social networks have attracted great interest in recentyears, largely on the account of their relevance to informa-tion processing in organizations, distributed search, anddiffusion of social influence. Networks evolve in time,driven by the shared activities and affiliations of their mem-bers; by similarity of individuals’ attributes; and by theclosure of short network cycles. I will discuss a study of adynamic social network comprising students, faculty, andstaff at a large university, in which interactions betweenindividuals are inferred from time-stamped e-mail headersrecorded over one academic year, and are matched withaffiliations and attributes. The results show that in theabsence of global perturbations, network-level propertiesappear stable, whereas individual properties fluctuate, andthe network evolution is driven by a combination of effectsarising from network topology itself and the organizationalstructure in which the network is embedded.

Gueorgi KossinetsCornell [email protected]

MS19

Persistent Homology of Leaf Networks

In this talk, I will present our attempt to get a better un-derstanding of various leaf networks by looking at theircertain geometric and topological descriptors. Our ap-proach is based on the notion of persistent homology. Ishall briefly describe the concept and show how it can beused to capture important information about leaf networksand perhaps lead to a novel method of leaf classification.

Yuriy MileykoGeorgia Institute of [email protected]

MS20

Capacitated Domination, Tetrominoes, and Com-puters

An r-capacitated dominating set of a graph G is a set{v1, . . . , vk} of vertices such that there is a partition(V1, . . . , Vk) of the vertex set where for all i, vi ∈ Vi, viis adjacent to all of Vi − {vi}, and |Vi| ≤ r + 1. Ther-capacitated domination number ¶r(G) is the minimumcardinality of an r-capacitated dominating set. We providebounds, properties and algorithmic results on ¶r. Some ofthe bounds are produce by an automated computer pro-gram. We also show a connection with tetrominoes, whichare polyominoes formed from four squares stuck togetherto form a T. We show that an n × n square with n oddcannot be tiled with tetrominoes to leave one hole. Thisincludes joint work with James Huff and Steve Hedetniemi.

Wayne Goddard

52 DM08 Abstracts

School of ComputingClemson [email protected]

MS20

A New Bound on the Domination Number ofGraphs with Minimum Degree Two

For a graph G, let γ(G) denotes the domination numberof G and let δ(G) denote the minimum degree among thevertices of G. A vertex x is called a bad-cut-vertex of Gif G − x contains a component, Cx, which is an induced4-cycle and x is adjacent to at least one but at most threevertices on Cx. A cycle C is called a special-cycle if C is a5-cycle in G such that if u and v are consecutive verticeson C, then at least one of u and v has degree 2 in G.We let bc(G) denote the number of bad-cut-vertices in G,and sc(G) the maximum number of vertex disjoint special-cycles in G that contain no bad-cut-vertices. We say that agraph is (C4, C5)-free if it has no induced 4-cycle or 5-cycle.Bruce Reed [Paths, stars and the number three. Combin.Probab. Comput. 5 (1996), 277–295] showed that if G isa graph of order n with δ(G) ≥ 3, then γ(G) ≤ 3n/8. Inthis paper, we relax the minimum degree condition fromthree to two. Let G be a connected graph of order n ≥ 14with δ(G) ≥ 2. As an application of Reed’s result, we showthat γ(G) ≤ 1

8(3n + sc(G) + bc(G)). As a consequence of

this result, we have that (i) γ(G) ≤ 2n/5; (ii) if G containsno special-cycle and no bad-cut-vertex, then γ(G) ≤ 3n/8;(iii) if G is (C4, C5)-free, then γ(G) ≤ 3n/8; (iv) if G is2-connected and dG(u) + dG(v) ≥ 5 for every two adjacentvertices u and v, then γ(G) ≤ 3n/8. All bounds are sharp.

Michael A. HenningUniversity of [email protected]

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

Anders YeoUniversity of London, United [email protected]

MS20

Protection of Graphs

In graph protection, guards located at vertices of a graph Gdefend the vertices against a sequence of attacks. A guardcan protect the vertex at which its located and move toa neighbouring vertex to defend an attack there. I shallmainly discuss eternal total domination, in which the se-quence of attacks is infinitely long and the configurationof guards induces a total dominating set before and aftereach attack has been defended.

Kieka MynhardtUniversity of VictoriaVictoria, BC, [email protected]

Chip KlostermeyerUniversity of North [email protected]

Paul GroblerStellenbosch UniversitySouth [email protected]

MS20

Random Procedures for Constructing DominatingSets


Dieter RautenbachInstitut fuer MathematikTU [email protected]

MS20

An Upper Bound on the Domination Number ofn-Vertex Connected Graphs


Burak StodolskyDepartment of MathematicsUniversity of Illinois Urbana [email protected]

MS21

(7,2)-Edge-Choosability of Some 3-Regular Graphs

A graph is (7, 2)-edge-choosable if, for every assignment oflists of size 7 to the edges, it is possible to choose two colorsfor each edge from its list so that no color is chosen fortwo incident edges. We show that every 3-edge-colorablegraph is (7, 2)-edge-choosable and also that many non-3-edge-colorable 3-regular graphs are (7, 2)-edge-choosable.

Daniel W. CranstonDIMACS [email protected]


MS21

Edge Colouring Cubic Toroidal Maps with TwoOdd Faces

We show that every 3-regular toroidal graph with atmost two odd faces and facewidth at least two is 3-edgecolourable. This partially answers a question of Grun-baum. Along the way, we characterize the family of suchgraphs having no face of length less than five.

Luis GoddynSimon Fraser University, Canada

DM08 Abstracts 53

[email protected]

Maryam Verdian RiziSimon Fraser [email protected]

MS21

Minimum Cost Colourings and Homomorphisms

I will survey recent results on various aspects of listhomomorphism and minimum cost homomorphism prob-lems; this will include work with and/or of Rafiey, Gupta,Karimi, Feder, Huang, Gutin, and Yeo. Relations withminimum cost colourings will be briefly mentioned.

Pavol HellSchool of Computer ScienceSimon Fraser [email protected]

MS21

Shortest Cycle Cover Problem for Graphs withMinimum Degree Three

The notion of flows in graphs is dual to that of graph col-oring. The Shortest Cycle Cover Conjecture, known toimply the Cycle Double Cover Conjecture (closely relatedto flows in graphs), asserts that every bridgeless graph Ghas a cover by cycles with total length at most 7m/5, wherem = |E(G)|. The best known general bound is 5m/3, dueto Alon and Tarsi (1985) and to Bermond, Jackson, andJaeger (1983). For cubic graphs, the best known bound of44m/27 is due to Fan (1994). For cubic graphs, we improvethe bound to 34m/21, and we extend the bound of 44m/27to graphs with minimum degree 3.

Tomas KaiserUniversity of West Bohemia, [email protected]

Daniel KralCharles [email protected]

Bernard LidickyCharles University, [email protected]

Pavel NejedlyDepartment of Applied MathematicsCharles University, [email protected]

Robert SamalSimon Fraser University, [email protected]

MS21

Cycles in Sparse Graphs

Erdos and Hajnal conjectured that in a graph of infinitechromatic number, the sum of reciprocals of cycle lengths

is divergent. We show that if an infinite increasing se-quence of even integers grows no faster than towers of two

(2, 22, 222

, . . .), then every n-vertex graph of average de-gree at least exp(8 log∗n) contains a cycle of length in thatsequence. This result is tight, in the sense that there aresequences growing slightly faster than towers of two forwhich the conclusion is no longer true.

Jacques VerstraeteUniversity of California-San [email protected]

MS22

There are 1,132,835,421,602,062,347 Nonisomor-phic One-Factorizations of K14

We have recently estab-lished by means of a computer search that a complete graphon 14 vertices has 98,758,655,816,833,727,741,338,583,040distinct and 1,132,835,421,602,062,347 nonisomorphic one-factorizations. This talk will give an overview of the tech-niques used in the search and the consistency checks carriedout to gain confidence in the results. A preprint is availableat (http://arxiv.org/abs/0801.0202).

Petteri KaskiHelsinki Institute for Information Technology HIITCS Dept., University of [email protected]

Patric OstergardHelsinki University of TechnologyDept. of Communications and [email protected]

MS22

On the Football Pool Problem

The Football Pool Problem amounts to determine thesmallest covering code of radius d of ternary words of lengthv. For d = 1 and v = 6, the optimal code size z6 is onlyknown to satisfy 65 ≤ z6 ≤ 73. Using integer program-ming techniques, isomorphism pruning and grid comput-ing (built using Condor and the grid-computing toolkit),we were able to show that z6 ≥ 71 using over 140 CPUyears delivered in roughly 92 days.

Jeff LinderothDept. of Industrial and Sys. Eng.University of Wisconsin, [email protected]

Francois MargotCarnegie Mellon UniversityTepper School of [email protected]

Greg ThainComputer Sciences Dept.University of Wisconsin, [email protected]

54 DM08 Abstracts

MS22

Covering Arrays Avoiding Forbidden Configura-tions

Covering arrays are combinatorial designs that are usedfor testing systems such as software, circuits and networks,where failures can be caused by the interaction betweentheir component/parameter values. In this talk, we con-sider designing covering arrays when certain pairs of pa-rameter values are prohibited from our test suites (e.g.invalid input combinations, explosive substance combina-tions). We show that the problem is NP-complete for gen-eral alphabet size g, while for g = 2, we characterize feasi-ble instances and discuss efficient algorithms.

Peter DanzigerRyerson [email protected]

Eric MendelsohnUniversity of TorontoDepartment of [email protected]

Lucia MouraUniversity of OttawaInformation [email protected]

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

MS22

Transversals in Squares

Define L(m,n) to be the greatest integer such that if eachsymbol in an m× n array appears at most L(m,n) times,then the array must have a transversal. There are a fewresults on L(m,n) and almost nothing is known about thesquare case. We here devise algorithms for the square caseand investigate extremal squares. The results apply toLatin squares, frequency squares, one factorizations of thecomplete graph and the football pool problem.

John van ReesUniversity of [email protected]

MS23

Graph Theoretic Methods for Identifying Redun-dancy in Protein-Protein Interaction Networks

We show how a simple ”folk” theorem in graph theory(probably due to Erdos) can help us identify simple sub-graphs that correspond to possible examples of redundancyin the protein-protein interaction network in yeast. Thisis joint work with Arthur Brady, Kyle Maxwell, and NoahDaniels.

Lenore CowenDept. of Computer ScienceTufts University

[email protected]

MS23

Combinatorics of RNA Secondary Structures

Under a suitable abstraction, complex biological problemscan reveal surprising mathematical structure. ModelingRNA folding by plane trees, we prove combinatorial the-orems yielding insight into the structure and function oflarge RNA molecules. We also, with an appropriate lo-cal move, obtain a graph of RNA configurations which isisomorphic to the lattice of noncrossing partitions. Thus,the interaction between discrete mathematics and molecu-lar biology motivates new combinatorial theorems as wellas advancing biological applications.

Christine E. HeitschSchool of MathematicsGeorgia [email protected]

MS23

Using Integer Linear Programming for Genome-Scale Phylogenetics

Phylogenetics (evolutionary tree inference) is an old prob-lem that has found new importance in the genomic era. Weuse integer linear programming to solve a multi-commodityflow formulation of binary maximum parsimony Steinertree inference, a variant particularly relevant to analyzinggenetic variations between individuals in a species. Thisapproach let us construct local phylogenies spanning thehuman genome, which we are using to make inferencesabout human history and the evolution of our genome.

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

Srinath SridharComputer Science DepartmentCarnegie Mellon [email protected]

Fumei LamComputer Science DepartmentBrown [email protected]

Guy BlellochComputer Science DepartmentCarnegie Mellon [email protected]

R. RaviTepper School of [email protected]

MS23

Distribution of Segment Lengths in Genome Rear-

DM08 Abstracts 55

rangements

In 1938, Dobzhansky and Sturtevant introduced the studyof gene orders for constructing phylogenetic trees. We usecombinatorial methods (permutations, generating func-tions, asymptotics, recursions, and enumeration formu-las) to study the distributions of the number and lengthsof conserved segments of genes between unichromosomalgenomes. - Hide quoted text - This generalizes classicalwork on permutations from the 1940s-60s by Wolfowitz,Kaplansky, Riordan, Abramson, and Moser, who studieddecompositions of permutations into strips of ascending ordescending consecutive numbers.

Glenn TeslerUniversity of California, San [email protected]

MS23

Graph Algorithms for Protein Structure Predictionand Analysis

Graph-based computational models are a powerful, flexi-ble and efficient way of modeling many biological systems.This talk presents a couple of graph models for proteinstructure prediction and analysis. In these models, a ver-tex corresponds to a residue and an edge indicates interac-tion/spatial proximity relationships between two residuesin a protein. Using these models, we can develop algo-rithms for the solutions of several challenging structurebioinformatics problems by exploring the special proper-ties of these graphs. These algorithms have time complex-ity dependent on the treewidth of a protein graph model,which is typically small for many protein structure relatedcomputational problems.

Jinbo XuToyota Technological Institute at [email protected]

MS24

Phase Transition in Random Graphs with DegreeConstraints

In this talk we discuss evolution and phase transition inrandom graphs with degree constraints. Of particular in-terest are random graphs with a given degree sequence.

Mihyun KangHumbollt University (Berlin)[email protected]

MS24

Avoiding Small Subgraphs in Achlioptas Processes

Consider the following generalization of the classical ran-dom graph process, known as the Achlioptas process. Ateach round, instead of receiving a single random edge, onereceives two random edges and chooses one of them in adeterministic, online fashion. We study the problem ofavoiding a fixed subgraph in this context, and determinethresholds for the avoidance of all cycles, cliques, and com-

plete bipartite graphs Kt,t.

Po-Shen LohDepartment of MathematicsPrinceton [email protected]

Michael KrivelevichDept. of MathematicsTel Aviv [email protected]

Benny SudakovDepartment of [email protected]

MS24

The Second Largest Component in the Supercriti-cal 2D Hamming Graph

The 2-dimensional Hamming graph H(2, n) consists of then2 vertices (i, j), 1 ≤ i, j ≤ n, two vertices being adjacentwhen they share a common coordinate. We examine ran-dom subgraphs of H(2, n) in percolation with edge proba-bility p, so that the average degree 2(n− 1)p = 1 + ε. Pre-vious work of van der Hofstad and Luczak had shown thatin the barely supercritical region n−2/3 ln1/3 n � ε � 1the largest component has size ∼ 2εn. Here we show thatthe second largest component has size close to ε−2, so thatthe dominant component has emerged. This is joint workwith Joel Spencer.

Malwina LuczakLSE (London)[email protected]

MS24

Properties of Random Graphs via Boltzmann Sam-plers


Konstantinos PanagiotouDepartement of Computer ScienceETH [email protected]

MS24

The Erdos Renyi Phase Transition

The Erdos-Renyi Phase transition is very well studied. Therandom graph G(n,p) has a phase transition when pt isnear one. In this semi-expository talk we describe how thetransition occurs and the appropriate mean field scaling sothat we can see the transition close up.

Joel SpencerDepartment of Mathematics, Courant InstituteNew York University, New York, NY 10012, [email protected]

56 DM08 Abstracts

MS25

Cryptography on Real Hyperelliptic Curves

An alternative to the traditional ”imaginary” model of ahyperelliptic curve is the less familiar ”real” model. Un-fortunately, the real model renders arithmetic in the Jaco-bian via reduced divisors less efficient. However, this modelexhibits another ”almost-group” structure, termed infras-tructure, which in addition to divisor composition supportsa second, much faster operation. This talk discusses thereal model of a hyperelliptic curve, its two-fold baby stepgiant step divisor arithmetic, and potential cryptographicapplications.

Renate ScheidlerUniversity of [email protected]

MS25

Complex Double Bases Applied to Scalar Multipli-cation on Algebraic Curves

In elliptic curve cryptography, the costliest operation isthe computation of nP = P + ... + P, called scalar mul-tiplication. It is acknowledged that the existence of fastendomorphisms (such as the Frobenius on Koblitz curves)results in clear performance speedup. I will expose howthe use of double base expansions of n gives way to a newclass of scalar multiplication algorithms capable of beatingthe fastest known implementations on Koblitz curves, withnegligible additional memory.

Francesco SicaMathematics & Computer ScienceMount Allison [email protected]

MS25

Non-Abelian Constructions in Cryptography:Challenges and Hopes

A number of attempts have been made to base securecryptographic schemes on computational problems in non-abelian algebraic structures. Proposals made range fromhash functions and public key encryption schemes to key es-tablishment protocols. Especially constructions using non-abelian groups received a lot of attention. Repeated crypt-analytic successes decreased the initial optimism signifi-cantly, however. With a focus on key establishment andhash functions, the talk surveys several non-abelian pro-posals that have been made in the last years and discussestheir security.

Rainer SteinwandtFlorida Atlantic [email protected]

MS25

Bounds for Generalized Separating Hash Families

Perfect hash families and separating hash families are com-binatorial structures that have numerous applications incryptography. In this talk, we look at generalized separat-

ing hash families and discuss some recent bounds (neces-sary conditions for existence) that have been proven. Theproof techniques are fairly elementary, and combinatorialin nature. The resulting bounds are quite close to the ex-istence results that can be obtained by application of theprobabilistic method.

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

MS25

A Fundamental Cryptographic (?) Algorithm

Cryptosystems which rely for their security on the pre-sumed difficulty of solving the discrete logarithm problemin quadratic number fields execute somewhat more slowlythan the standard Diffie-Hellman or RSA techniques. Al-though this gap has narrowed somewhat in the last severalyears, in order to narrow it further, there is still a funda-mental problem that must be addressed. In this talk, I willdescribe the most recent progress on this problem.

Hugh WilliamsDepartment of Mathematics & StatisticsUniversity of [email protected]

MS26

Ranking Groups by Their Actions

We consider ways to extend the definition of the indegreeand outdegree of a vertex to a set of vertices in an edge-weighted directed graph. This measure serves as a modelfor ranking groups based on how they act on or interactwith others. The measure itself and related statistics in-cluding its mean and variance can be computed quickly andhence can be used for a variety of applications that requireeither scalability or real-time, interactive data analysis.We pay special attention to applications to biology, mar-ket basket analysis, and criminal network analysis. Key-words: indegree, outdegree, edge weights, directed graph,data analysis

Nathaniel DeanTexas State University, San [email protected]

Kiran ChilakamarriTexas Southern [email protected]

MS26

Digraph Irregularity Strength

It is an elementary exercise to show that any non-trivialsimple graph has two vertices with the same degree. Thisis not the case for digraphs and multigraphs. We considergenerating irregular digraphs from arbitrary digraphs byadding multiple arcs. To this end, we define an irregularlabeling of a digraph D to be an arc labeling of the digraphsuch that the ordered pairs of the sums of the in-labels and

DM08 Abstracts 57

out-labels at each vertex are all distinct. We define thestrength s(D) of D to be the smallest of the maximum la-bels used across all irregular labelings. A similar definitionfor graphs has been studied extensively In this talk, we givea general lower bound on s(D) and determine s(D) exactlyfor tournaments, directed paths and cycles and the orienta-tion of the path where all vertices have either in-degree 0 orout-degree 0. We also determine the irregularity strengthof a union of directed cycles and a union of directed paths,the latter which requires a new result pertaining to find-ing circuits of given lengths containing prescribed verticesin the complete symmetric digraph with loops. Keywords:Digraph, Irregularity Strength, Irregular Labeling.

Michael FerraraDept. of Theoretical and Applied MathematicsUniversity of [email protected]

Jesse Gilbert, Mike JacobsonThe University of Colorado, [email protected],[email protected]

MS26

An Extension of Strongly Regular Graphs

The Friendship theorem states that if any two people in aparty have exactly one common friend, then there exists apolitician who is everybodys friend. In this talk we gener-alize the Friendship theorem. Let λ be any non-negativeinteger and μ be any positive integer. Suppose each pairof friends has exactly λ common friends, and each pair ofstrangers have exactly μ common friends in a party. (Thecorresponding graph is a generalization of strongly regulargraphs by relaxing the regularity property on vertex de-grees.) We prove that either everyone has exact the samenumber of friends or there exists a politician who is every-bodys friend. Key Words: strongly regular graphs, openneighborhoods

Ralucca M. GeraNaval Postgraduate [email protected]

Jian ShenTexas State University, San Marcosemail: [email protected]

MS26

Computer Methods for Finding Graph Decompo-sitions

Integer programming is a powerful technique for modelingand solving combinatorial search problems. However, thecommon methods of solving integer programs fare poorlywhen there is a large amount of symmetry present. We willdiscuss the use of canonical labellings to reduce the searchspace for the particular application of finding regular graphdecompositions.

Stephen G. HartkeUniversity of [email protected]

MS26

The Rainbow Connectivity of Regular Graphs

A path P in an edge-colored graph (not necessarily a properedge-coloring) is a rainbow path if no two edges are as-signed the same color. For a connected graph G with con-nectivity κ(G) and an integer k with 1 ≤ k ≤ κ(G), therainbow k-connectivity rck(G) of G is the minimum num-ber of colors needed in an edge-coloring of G such that ev-ery two distinct vertices u and v of G are connected by atleast k internally disjoint u - v rainbow paths. We presentsome results and open questions in this area of research.

Futaba OkamotoUniversity of Wisconsin - La [email protected]

PP0

Greed Is Good, But Branching Is Better: A Local-Ratio Style Improvement for Hitting Set of Bundles

In this work, we extend the approximation analysis of theHitting Set of Bundles (HSB) problem using a greedy,rather than LP-based approach. Using this combinatorialframework, we derive a theorem regarding the structure ofthe worst-case instances of HSB that leads us to a sim-ple branching improvement scheme. We prove that thisbranching approach yields the best approximation ratio onan important class of HSB problems, and present experi-mental results for more general instances.

Kevin M. Byrnes, Alan GoldmanDepartment of Applied MathematicsJohns Hopkins [email protected], [email protected]

PP0

A Tighter Linear Programming Relaxation of Dis-joint Combinatorial Rectangle Cover and Analysisof Its Solution Space

Disjoint combinatorial rectangle cover is an importantproblem because it presents a lower bound on communi-cation complexity. It can be written in the form of integerprogramming problem. Karchmer, Kushilevitz and Nisanshowed a technique to prove a lower bound on this inte-ger programming problem by LP relaxation. To advancethis technique we proposed a tighter LP relaxation of dis-joint combinatorial rectangle cover problem and analyzedits solution space using spectral graph theory.

Norie Fu, Kenya UenoThe University of Tokyof [email protected], [email protected]

PP0

The Partition-Regularity of Pythagorean Triples

The problem at the heart of this research involves under-standing the partition-regularity of Pythagorean triples. Ifone colors the natural numbers with k-colors, must therebe a monochromatic solution to the equation a2 + b2 =c2? Our approach to the problem is two-fold, including

58 DM08 Abstracts

mathematical analysis of the Pythagorean triples as wellas creating software to generate bounds for different color-ing schemes.

Christopher PoirelUniversity of South [email protected]

Joshua CooperUniversity of South CarolinaDepartment of [email protected]

dm08 abstracts - society for industrial and applied ... · dm08 abstracts 29 michael o. albertson...

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