7th c5 graph theory workshop - mathe.tu-freiberg.de file7th c5 graph theory workshop ”cycles,...
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7th C5 GRAPH THEORY WORKSHOP
”Cycles, Colorings, Cliques, Claws and Closures”
Kurort Rathen, 12.-16.05.2003
SELECTED PROBLEMS
http://www.mathe.tu-freiberg.de/
math/inst/theomath/WorkshopRathen.html
COLORINGS
1 Rainbow Cycles
(Ingo Schiermeyer)
For n ≥ k ≥ 3, let f(n, Ck) denote the maximum m for which it is
possible to color the edges of the complete graph Kn with m colors in
such a way that each k−cycle Ck in Kn has at least two edges of the
same color. Hence, for ≥ 1 + f(n, Ck) colors, Kn always contains a
multicolored Ck, also called Rainbow Cycle Ck.
Conjecture 1.1 (Erdos, Simonovits, Sos)
f(n, Ck) = n
(k − 2
2+
1
k − 1
)
+ O(1)
Known:
1. f(n, C3) = n − 1 (Erdos, Simonovits, Sos)
2. f(n, C4) = n +⌊
n3
⌋− 1 (Alon)
3. true for k = 5, 6, 7 (Jiang, Schiermeyer, West)
2 3-Colorability for Pk−free Graphs
(Bert Randerath, Ingo Schiermeyer, Meike Tewes)
Let Pk be the induced path on k vertices and 3−COL(Pk−free) be the
3−colorability problem for the class of Pk−free graphs.
Question 2.1 Does there exist an integer k ≥ 7 such that 3 −COL(Pk−free) remains NP-complete?
Known:
1. 3 − COL is NP-complete
2. 3 − COL(Pk−free) for k = 5, 6 is decidable in polynomial time
1
COLORINGS
3 On Cyclic Chromatic Number of3-connected Plane Graphs
(Mirko Hornak)
The cyclic chromatic number of a plane graph G, in symbol χc(G), is
a minimum number of colors in such a vertex coloring of G that distinct
vertices incident with a common face receive distinct colors. Clearly,
χc(G) ≥ ∆∗(G), where ∆∗(G) is the maximum face degree of G. On
the other hand, no 3-connected plane graph G is known with χc(G) >
∆∗(G) + 2. Plummer and Toft [6] proved that χc(G) ≤ ∆∗(G) + 9
and conjectured [PTC] that χc(G) ≤ ∆∗(G) + 2 for any 3-connected
plane graph G. It is known that PTC is true for ∆∗(G) = 3 (Four
Color Theorem), ∆∗(G) = 4 (Borodin [1]) and ∆∗(G) ≥ 24 (Hornak and
Jendrol’ [5]). For ∆∗(G) ≥ 60, by Enomoto, Hornak and Jendrol’ [4],
even an absolute result holds: χc(G) ≤ ∆∗(G) + 1 (graphs of pyramids
show that the bound ∆∗(G) + 1 cannot be improved). A best general
upper bound so far is due to Enomoto and Hornak [3], namely χc(G) ≤∆∗(G) + 5. For ∆∗(G) = 5, Borodin, Sanders and Zhao [2] succeeded to
show that χc(G) ≤ ∆∗(G) + 3.
Problem 3.1 Tackle PTC for ∆∗(G) = 5.
References:
[1] O. V. Borodin, Solution of Ringel’s problem on vertex-face coloring of plane graphs and coloring of 1-planargraphs (in Russian)Met. Diskr. Anal. 41 (1984), 12-26
[2] O. V. Borodin, D. P. Sanders, Y. Zhao, On cyclic colorings and their generalizationsDiscrete Math. 203 (1999), 23-40
[3] H. Enomoto, M. Hornak, A general upper bound for the cyclic chromatic number of 3-connected plane graphsmanuscript
[4] H. Enomoto, Hornak and S. Jendrol’, Cyclic chromatic number of 3-connected plane graphsSIAM J. Discrete Math., to appear
[5] M. Hornak and S. Jendrol’, On a conjecture by Plummer and ToftJ. Graph Theory 30 (1999), 177-189
[6] M. D. Plummer and B. Toft, Cyclic coloration of 3-polytopesJ. Graph Theory 11 (1987), 507-515
2
COLORINGS
4 The Circular Total Chromatic Number
(Andrea Hackmann, Arnfried Kemnitz)
A k−total coloring of a simple graph G is an assignment of k colors
to the vertices and edges of G such that the neighbored elements - two
adjacent vertices or two adjacent edges or a vertex incident to an edge
- are colored differently. The minimum number k for which a graph G
admits a k−total coloring is the total chromatic number χ′′(G) of G.
If k and d are positive integers with k ≥ 2d then a (k, d)−total coloring of
a graph G is an assignment c of colors {0, 1, . . . , k−1} to the vertices and
edges of G such that d ≤ |c(xi)− c(xj)| ≤ k− d whenever two elements
xi and xj are neighbored. The circular total chromatic number χ′′c (G)
of G is defined as the infimum of fractions kd
for all (k, d)−total colorings
of G:
χ′′c (G) = inf
{k
d: G has a (k, d) − total coloring.
}
Obviously, a (k, 1)−total coloring is a k−total coloring of G which im-
plies that χ′′c (G) ≤ χ′′(G). For cycles Cp it holds χ′′
c (C3k+1) = 3 + 1k
and
χ′′c (C3k+2) = 3 + 1
2k+1 whereas χ′′(C3k+1) = χ′′(C3k+2) = 4.
For example, for complete graphs and several classes of complete multi-
partite graphs the total chromatic number and the circular total chro-
matic number coincide.
Problem 4.1 Determine classes of graphs G aside from cycles such
that
χ′′c (G) < χ′′(G).
3
COLORINGS
5 List Colorings of Integer Distance Graphs
(Arnfried Kemnitz)
Let D be a subset of the positive integers IN. The integer distance graph
G(ZZ, D) = G(D) is defined as the graph with the set of integers as ver-
tex set, V (G(D)) = ZZ, and edge set consisting of all pairs uv whose
distance |u − v| is an element of the so-called distance set D.
General bounds for the chromatic number of integer distance graphs are
2 ≤ χ(G(D)) ≤ |D| + 1.
Voigt (1999) and Zhu (1996) determined χ(G(D)) if |D| = 3 :
If D = {x, y, z} consists of integers whose greatest common divisor
equals 1, then χ(D) = 4 if and only if D = {1, 2, 3n} or D = {x, y, x+y}and x 6≡ y (mod 3). If x, y, z are odd then χ(D) = 2. For all other
3−element distance sets D it holds χ(D) = 3.
General bounds for the list chromatic number (choice number) of integer
distance graphs are χ(D) ≤ ch(D) ≤ |D|+1 (Kemnitz, Marangio 2001).
Question 5.1 Does there exist a 3−element distance set such that
ch(D) < 4?
References:
[1] A. Kemnitz, M. Marangio, Edge colorings and total colorings of integer distance graphsDiscussiones Mathematicae Graph Theory, to appear
[2] M. Voigt, Colouring of distance graphsArs Combinatoria 52 (1999), 3-12
[3] X. Zhu, Distance graphs on the real linemanuscript, 1996
4
COLORINGS
6 Choice Number of Cartesian Products
(Mieczyslaw Borowiecki, Stanislav Jendrol’)
Let ch(G) denote the choice number of G and let G×H be the Cartesian
Product of graphs G and H . Galvin (1995) proved that ch(Kn×Kn) = n
and solved in this way the old Dinitz’s conjecture. (See Diestel’s book
where the solution is presented as a result on the edge list coloring of
bipartite multigraphs.)
Let G and H be graphs. Clearly,
max{ch(G), ch(H)} ≤ ch(G × H).
Question 6.1 Does there is an absolute constant c such that
ch(G × H) ≤ max{ch(G), ch(H)} + c?
Question 6.2 If the answer is YES, then how big is c? Is c = 1?
5
COLORINGS
7 Coloring the Square of a Graph
(Stanislav Jendrol’)
Conjecture 7.1 Let G be a planar graph with δ(G) ≥ 5, and let G2
be the square of G. Then for the chromatic number of G2 it holds
χ(G2) ≤ ∆(G) + 25
8 Fruit Salad
(Andras Gyarfas)
Conjecture 8.1 If each path of a graph G spans a 3−colorable sub-
graph, then G is k−colorable with a constant k (perhaps with k = 4).
Comment: k = 4 would be best possible.
Known: These graphs are colorable with 3 · blgc |V (G)|c colors for a
suitable constant c = 87 (cf. [1]).
References:
[1] B. Randerath, I. Schiermeyer, Chromatic Number of Graphs each Path of which is 3-colourableResult. Math 41 (2002), 150-155
6
COLORINGS
9 Non-Repetitive Colorings of Graphs
(JarosÃlaw Grytczuk, Mariusz HaÃluszczak)
The problem we consider has emerged as a graph theoretical variant of
the famous non-repetitive sequences of Thue. A finite sequence a =
a1a2 . . . an of symbols from a finite set S is called non-repetitive if it
does not contain a sequence of the form xx = x1x2 . . . xmx1x2 . . . xm
as a subsequence of consecutive terms. For instance, the sequence a =
123132123213 over the set of symbols S = {1, 2, 3} is non-repetitive,
while b = 1232321 is not. A beautiful theorem of Thue asserts that
there exist arbitrarily long non-repetitive sequences of only three differ-
ent symbols.
A coloring of the set of vertices of a given graph G is called non-repetitive
if a sequence of colors on any path of G is non-repetitive. The minimal
number of colors that do the job is denoted by π(G). So, for instance, the
theorem of Thue says that π(Pn) = 3, for all n ≥ 4, where Pn denotes
a path with n vertices. Obviously, π(G) is equal at least the chromatic
number of G.
Question 9.1 Is it true that π(G) ≤ c∆(G) for some absolute con-
stant c?
In case of a negative answer to the above question we propose a weaker
statement.
Question 9.2 For every positive integer k is there a constant ck
such that π(G) ≤ ck, for all graphs G with ∆(G) ≤ k?
And one example of a more concrete problem concerning cycles. By the
theorem of Thue one gets easily that π(Cn) ≤ 4, but we have found,
by a computer search, that π(Cn) = 3, for all n ≤ 1000, except n =
5, 7, 9, 10, 14, 17.
Question 9.3 Is it true that π(Cn) = 3, for all n ≥ 18?
7
CYCLES
10 k-1 Vertices on a Common Cycle
(Jochen Harant)
It is known that k prescribed vertices of a k−connected graph belong
to a common cycle. For arbitrary c > 0 there is a k−connected graph
G containing a longest cycle of length at least c (i.e. the circumference
of G is at least c) and a set X of k vertices of G such that the length
of an arbitrary cycle of G containing X is less than 2k + 1. In other
words: Although k prescribed vertices of a k−connected graph belong
to a common cycle it is impossible to guarantee a ”long” cycle through
these vertices. We believe that the situation changes if k − 1 prescribed
vertices of a k−connected graph are considered:
Question 10.1 Is it true that for any k ≥ 2 there is a constant
a (depending only on k) such that arbitrary k − 1 vertices of a
k−connected graph with circumference c belong to a common cycle
of length at least c2 + a?
Known: The answer is YES for k = 2 and k = 3.
11 l Vertices on a Short Cycle
(Jochen Harant)
Let G be a k−connected graph of order n = |V (G)|. Given l prescribed
vertices, 1 ≤ l ≤ k, find a short cycle containing these l vertices. By a
theorem of Dirac such a cycle always exists. Denote the length of this
cycle in the worst case by f(n, k, l).
Question 11.1
f(n, k, k) =2
kn + ck ?
8
CYCLES
Known:
1. f(n, k, 1) = 2
(k
2)n + const
2. f(n, k, 2) = f(n, k, 3) = 2kn + const
3. For l > k: If such a cycle exists, then
f(n, 3, 4) ≥3
4n + const
12 Prescribed Vertices and Edges on a Cycle
(Jochen Harant, Tobias Gerlach)
Given an integer k ≥ 2, let G be a k-connected graph, X ⊂ V (G),
Y ⊂ E(G−X), and Y be independent. It is known that there is a cycle
of G containing X ∪ Y if |X| + |Y | = k and |X| ≥ 1 or if k is even,
|Y | = k, and |X| = 0.
Problem 12.1 Find a ’good’ bound b such that a similar result holds
if |X| + |Y | = k + 1 and toughness(G) ≥ b !
Comment: If |X| ≥ 4 then results are known - the case |X| ≤ 3 is the
problem!
References:
[1] G.A. Dirac, 4-chromatische Graphen und vollstandige 4-GraphenMath. Nachr. 22 (1960), 51-60
[2] R. Haggkvist, C. Thomassen, Circuits through specified edgesDiscrete Math. 41 (1982), 29-34
[3] J. Harant, On paths and cycles through specified verticesaccepted in Discrete Math.
[4] K. Kawarabayashi, One or two disjoint Circuits cover independent edgesJ. Combin. Theory Ser. B 84 (2002), 1-44
[5] M.E. Watkins, D.M. Mesner, Cycles and connectivity in graphsCan J. Math. 19 (1967), 1319-1328
9
CYCLES
13 Vertex Disjoint Cycles
(Hikoe Enomoto, Stanislav Jendrol’)
(Ingo Schiermeyer)
Let f(k, α) := max{|V (G)| | α(G) ≤ α and G has no k vertex disjoint
cycles}, where α(G) is the independence number of G.
Problem 13.1 Find a pair k, α and (an explicit) graph G such that
|V (G)| > 2α + 3k − 3, α(G) ≤ α and G has no k disjoint cycles.
Known: (Enomoto, Jendrol’, Schiermeyer, Egawa, Ota)
1. f(k, α) ≥ 2α + 3k − 3
K3k−1 +
(α − 1)K2
︸ ︷︷ ︸
2. f(k, α) = 2α + 3k − 3, for
α = 1, 2, 3, 4, 5
k = 1, 2
k = 3 and δ ≥ 4
3. For any c > 0 there exist α, k and a graph G such that
f(k, α) > c(k + α(G))
References:
[1] Y. Egawa, H. Enomoto, S. Jendrol, K. Ota, I. Schiermeyer, Independence number and vertex disjoint cycles,preprint, 2002
10
CYCLES
14 Vertex-dominating Cycles
(Akira Saito)
A cycle C in a graph G is said to be a vertex-dominating cycle if V (C)
is a vertex-dominating set of G (i.e. every vertex in G has distance at
most one from C). Therefore a hamiltonian cycle is a vertex-dominating
cycle, but not all the vertex-dominating cycles are hamiltonian cycles. A
Japanese graph theorist, T. Yamashita, proposed the following.
Conjecture 14.1 (Yamashita)
Let G be a bipartite graph with partite sets V1 and V2. Let |V1| = n1,
|V2| = n2 and suppose n1 + 5 ≤ n2. If the minimum degree of G is
at least 13(n1 + 5), then G has a vertex-dominating cycle.
Theorem 14.1 (Yamashita)
Let G be a bipartite graph with partite sets V1 and V2. Let |V1| = n1,
|V2| = n2 and suppose n1 ≤ n2. If the minimum degree of G is at
least 13(n2 + 1), then G has a vertex-dominating cycle.
A vertex-dominating cycle is a simple generalization of a hamiltonian
cycle. However, unlike hamiltonian cycles, when we consider a vertex-
dominating cycle in a bipartite graph, we can drop the trivial necessary
condition that the graph is balanced, and I think it interesting. Ya-
mashita got a lower bound which guarantees the existence of a vertex-
dominating cycle. But it is a function of the order of the larger partite
set. In this sense any lower bound which is a function of the order of the
smaller partite set looks fine.
11
CYCLES
15 Nonpancylic claw-free graphs withcomplete closure
(Zdenek Ryjacek, Richard Schelp)
It is known that a claw-free graph G is hamiltonian if and only if its
closure cl(G) is hamiltonian. On the other hand, there are nonpancyclic
graphs with pancyclic closure [1]. The graph in the figure below is an ex-
ample of such a nonpancyclic graph with complete (and hence pancyclic)
closure.
•••••••••••
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Problem 15.1 Determine the maximum number of cycle lengths
that can be missing in a claw-free graph on n vertices with complete
closure.
Conjecture 15.1 Let c1, c2 be fixed constants. Then for large n,
any claw-free graph G of order n whose closure is complete contains
cycles Ci for all i, where 3 ≤ i ≤ c1 and n − c2 ≤ i ≤ n.
It is easy to see that a claw-free graph with complete closure on at least
4 vertices can miss neither a C3 nor a C4. The main result of [2] shows
that such a graph G cannot be missing a cycle of length n− 1; however,
the proof of this result is difficult and cannot be iterated.
References:
[1] Brandt, S.; Favaron, O.; Ryjacek, Z., Closure and stable hamiltonian properties in claw-free graphsJ. Graph Theory 34 (2000), 30-41
[2] Ryjacek, Z.; Saito, A.; Schelp, R.H., Claw-free graphs with complete closureDiscrete Mathematics (to appear)
12
CYCLES
16 Every locally connected graph is weaklypancyclic
(Zdenek Ryjacek)
Let G be a finite simple undirected graph and let g(G) and c(G) be the
girth and the circumference of G (i.e. the length of a shortest cycle of
G and the length of a longest cycle of G), respectively. We say that G
is weakly pancyclic if G contains cycles of all lengths ` for g(G) ≤ ` ≤c(G). The graph G is locally connected if the neighborhood of every
vertex of G induces a connected graph.
Conjecture 16.1 Every connected locally connected graph is weakly
pancyclic.
Known:
1. True for claw-free graphs (Clark [1] proved that every connected,
locally connected claw-free graph on at least three vertices is
vertex pancyclic).
2. True for planar triangulations (unpublished proof by P. Balister,
Memphis).
References:
[1] Clark, L., Hamiltonian properties of connected locally connected graphsCongr. Numer. 32 (1981), 199-204
13
INDEPENDENT SETS
17 Maximum Independent Sets in Graphswith Maximum Degree 3
(Ingo Schiermeyer)
Let G be a K4−free graph with maximum degree 3. Then the indepen-
dence number α(G) of G is bounded from below by
α(G) ≥∑
v∈V (G)
1
d(v) + 1≥
n
4(Caro − Wei bound).
On the other hand, by Brook’s theorem, G has chromatic number χ(G) ≤∆(G) ≤ 3 and therefore α(G) ≥ n
3 .
For triangle-free graphs of maximum degree 3 it is known that α(G) ≥514n, which is sharp for the generalized Petersen graph P (7, 2).
Problem 17.1 Find a good (sharp) general bound for the indepen-
dence number of K4−free graphs with maximum degree 3.
Comment: There are classes of graphs with maximum degree 3 and
α(G) = n3 , which contain n
6 triangles.
14
DECOMPOSITIONS & PARTITIONS
18 Path Partition Conjecture
(Ingo Schiermeyer)
The length of a longest path in a graph G is denoted by τ (G). A partition
of the vertex set of G such that τ (G[A]) ≤ a and τ (G[B]) ≤ b is called
an (a, b)−partition of G. If G has an (a, b)−partition for every pair (a, b)
of positive integers such that a + b = τ (G), then G is τ−partitionable.
Conjecture 18.1 [PPC] Every graph is τ−partitionable.
The PPC holds for all graphs G with n − 1 ≤ τ (G) ≤ n. Recently we
proved it for n − 3 ≤ τ (G) ≤ n − 2.
Question 18.1 Does the PPC hold for τ (G) = n − 4?
Known: asymptotic result (Frick, Schiermeyer)
For a graph G with τ (G) = n − p for some fixed p ≥ 4 the PPC holds
provided that n ≥ p(10p − 3).
15
DECOMPOSITIONS & PARTITIONS
19 Arbitrarily Vertex-decomposable Trees
(Mirko Hornak and Mariusz Wozniak)
A tree T is arbitrarily vertex-decomposable (avd for short) if for any
sequence (t1, . . . , tk) of positive integers satisfying∑k
i=1 ti = |V (T )|there is a sequence (T1, . . . , Tk) of subtrees of T such that {V (Ti) :
i ∈ {1, . . . , k}} is a decomposition of V (T ) and |V (Ti)| = ti for each
i ∈ {1, . . . , k}. A star-like tree is a tree homeomorphic to K1,q, q ≥ 3,
whose arms (subtrees with endvertices of degrees 1 and q) are of or-
ders a1, . . . , aq; such a tree is denoted by S(a1, . . . , aq). Without loss of
generality we may suppose that (a1, . . . , aq) is a non-decreasing sequence.
Trivially, paths are avd. We know that S(2, a2, a3) is avd if and only if
a2 and a3 are coprime. Moreover, there are sequences (a1, a2, a3) with
a1 ≥ 3 and sequences (2, a2, a3, a4) such that the corresponding star-like
trees are avd. On the other hand, a sequence A = (a1, . . . , aq) is not
avd whenever one of the following assumptions is satisfied: (i) q ≥ 7; (ii)
q = 6 and a1 ≥ 3; (iii) q = 5 and a1 ≥ 5.
Conjecture 19.1 Let A = (a1, . . . , aq) be a non-decreasing sequence
of integers with a1 ≥ 2. Then the star-like tree S(A) is not arbitrarily
vertex-decomposable if either q ≥ 5 or q = 4 and a1 ≥ 3.
If our conjecture with q ≥ 5 is true, it can be shown that any avd tree is
of maximum degree at most 4. Note that for every ∆ ∈ {3, 4} there are
avd trees of maximum degree ∆ that are not star-like.
16
DECOMPOSITIONS & INTERSECTIONS
20 Conjectures on Decompositionswith Remainder or Surplus
(ZdzisÃlaw Skupien)
Let p and t be positive integers. Let
ρ =
(p
2
)
mod t, 0 ≤ ρ < t,
i.e., ρ is the remainder on dividing p(p − 1)/2 by t.
Conjecture 20.1 If ρ > 0 then, for each ρ-subset R ⊆ E(Kp) (with
|R| = ρ), the graph Kp − R is edge-decomposable into t isomorphic
subgraphs.
Conjecture 20.2 If ρ > 0 then, for each multiset S of edges of the
complete graph Kp of size |S| = t − ρ and with ∆(< S >) (= |S|/p,say) small enough, the multigraph Kp + S is decomposable into t
isomorphic submultigraphs.
Comments: If ρ = 0, both Conjectures 1 and 2 (with S = ∅) are true
due to results by Harary-Robinson-Wormald and Schonheim-Bialostocki,
both published in 1978. If the phrase “for each” in both Conjectures is re-
placed by “for a certain”, the resulting statements are proved by Skupien
[2].
Hence both Conjectures are true if t ≤ 2 or t ≥(p
2
). Moreover, Conjec-
tures 1 and 2 are true if |R| ≤ 1 (which is the case if t = 3) and |S| = 1,
respectively; also if decomposition parts are to be single edges (hence if
t >(p
2
)/2 in Conjecture 1). Furthermore, if any of remaining values of
t is fixed, each Conjecture can be reduced to a finite number of values
for p; for instance, only to some p ≤ 4t − 5 (e.g., p = 10, 11, 12, 18 if
t = 7 but if t = 5 then p = 8 only) in case of Conjecture 1, cf. [2]. Inde-
pendently, Conjecture 1 has been checked for p ≤ 8 whence for t = 5, too.
17
DECOMPOSITIONS & INTERSECTIONS
It can be noted now that Conjecture 1 is supported by the following deep
result of Plantholt ’81. Graphs G of odd order 2s + 1 with a spanning
star have chromatic index ∆ (= 2s) iff the complement G of G has any
s or some more edges of K2s+1 (so that δ(G) = 0). This result is clearly
equivalent to sK2 | K2s+1−R for any R ⊂ E(K2s+1) with |R| = s (then
t = 2s, p = 2s+1 and all decomposition parts are maximum matchings).
Relaxation: Consider scattered R (with ∆(R) → min) or clustered R
(with order of < R >→ min).
It is noted in [2] that Conjecture 2 can be false if R is not scattered.
References:
[1] M. Plantholt, The chromatic index of graphs with a spanning starJ. Graph Theory 5 (1981) 45-53
[2] Z. Skupien, The complete graph t-packings and t-coveringsGraphs Comb. 9 (1993) 353-363
21 Nonempty Intersections
(Stanislav Jendrol’, ZdzisÃlaw Skupien)
Conjecture 21.1 In a connected (2−connected) graph G any three
longest paths (cycles) have a nonempty intersection.
Known: In a connected (2−connected) graph any seven (nine) longest
paths (cycles) need not have a nonempty intersection.
References:
[1] S. Jendrol’, Z. Skupien, Exact numbers of longest cycles with empty intersectionEur. J. Comb. 18 No.5 (1997), 575-578
18
WEIGHT OF GRAPHS
22 Weight of Graphs havinga Given Property
(Stanislav Jendrol’)
By the weight w(e) of an edge e = xy of a graph G = (V, E) we mean
the degree sum of its endvertices x and y;
w(e) := degG(x) + degG(y).
For a graph G its weight is defined as follows
w(G) := min{w(e)|e ∈ E(G)}.
Denote by P a graph property. Let G(n, m,P) be a family of graphs with
n vertices, m edges and having the property P . Our general problem is
the following
Problem 22.1 Determine the value
W (n, m,P) = max{w(G)|G ∈ G(n, m,P)}.
Remarks:
1. For P being the family of all graphs the problem was prosed by P.
Erdos to the Prachatice Conference in 1990 and was completely
solved in [JS] where the reader can find more informations to the
related questions and problems.
2. For the property P to be a family of graphs of average degree α
the problem is investigated in [1].
3. The problem seems to be very interesting for the cases when Pis the family of connected graphs, bipartite graphs, or k-partite
graphs, ...
References:
[1] Bose P., Smid M., Wood D.R., Light edges in degree-constrained graphsProc. Canadian Conf. on Computational Geometry (CCCG ’02), 2002.
[2] Jendrol’ S., Schiermeyer I., On max-min problem concerning weight of edgesCombinatorica 21 (2001), 351-359
19
GRAPH EDITING
23 Graph Editing
(Maria Axenovich)
Consider a fixed graph H . For each graph G, let f(G, H) be the
minimal number of edge-deletions and edge-additions in G such that
the resulting graph does not contain H as an induced subgraph. Let
f(n, H) = max{f(G, H) : n(G) = n}.
This problem is motivated by the analogous notion on editing of se-
quences when one sequence is obtained from another by element deletion
or element addition. In graphs, the similar problem was studied by Erdos
when the only allowed operation is edge-deletion. Namely, it was investi-
gated how many edges in a triangle-free graph does one need to delete to
obtain a bipartite graph. The edition problem of graphs was also stud-
ied for some trees and has an important application in finding concensus
trees. Among many questions to answer are the following:
Question 23.1 What if f(n, H) for H is a specific class of graphs,
for instance with fixed chromatic number?
Question 23.2 If f(n, H) = f(G, H), is it true that we need both
edge-deletions and edge-additions to avoid an induced copy of H in
G by f(G, H) operations?
There are few partial results on this problem.
20
FACTORS, BINDING FUNCTIONS & CUTSETS
24 2-Factor with k Components
(Ralph Faudree, Ronald Gould, Mike Jacobson)
Theorem 24.1 (Dirac)
If δ(G) ≥ n2 then G is hamiltonian (n = |V (G)|).
Question 24.1 Does there exist a constant c such that if G is hamil-
tonian and δ(G) ≥ c · n, then G has a 2−factor with exactly k com-
ponents?
Known:
1. The question is proved for c = 512. (Faudree Gould, Jacobson)
2. δ = 4, k = 2 6⇒ 2−factor with 2 components
K5 − eK5 − e
Conjecture 24.1 (Gould) δ(G) ≥ const · log n suffices
25 χ−Binding Functions
(Bert Randerath, Ingo Schiermeyer)
Problem 25.1 Determine a χ−binding function f(w) for the class
GI(3, 4), where GI(n1, n2, . . . , nk) is the class of all graphs whose
induced cycle lengths are one of n1, n2, . . . , nk.
21
FACTORS, BINDING FUNCTIONS & CUTSETS
Theorem 25.1 (Randerath, Schiermeyer, Hoang, McDiarmid,
1999) There exists no linear χ−binding function for GI(3, 4).
Example:
G1 = C7, Gi+1 = Gi[ C7 ]
n(Gi) = 7i, α(Gi) = 2i, ω(Gi) = 3i
χ(Gi) ≥(
72
)i χ
ω(Gi) ≥
(76
)i
26 Acyclic Cutsets
(Atsushi Kaneko)
Conjecture 26.1 Let G be a graph of order n. If |E(G)| ≤ 3n − 7,
then G has an acyclic cutset.
Comment: The upper bound is sharp, K3 ∨ (n − 3)K1:
Known:
1. Every graph on n vertices and at most 2n− 4 edges contains an
independent vertex-cut (cf. [1]). For sharpness see e.g. Kn−2 +
K2.
2. The conjecture is true for planar graphs. (a corollary of a result
in [2]).
References:
[1] G. Chen, X. Yu, A note on fragile graphsDiscrete Math. 249 No. 1-3 (2002), 41-43
[2] M. Aigner, Graphentheorie - Eine Entwicklung aus dem 4-FarbenproblemMonographie
22
FACTORS, BINDING FUNCTIONS & CUTSETS
27 Stable Cutsets
(Bert Randerath)
STABLE CUTSET: Given a graph G. Does G contain a stable cutset?
Problem 27.1 Is STABLE CUTSET in P or NP-complete on graphs
with maximum degree 4?
Comments: STABLE CUTSET is in P for graphs with maximum
degree 3 (see [1],[2]), for line graphs of maximum degree 4 (see [1]), and
for graphs with n vertices and at most 2n − 4 edges ([2]). STABLE
CUTSET is in NP-c for 5−regular line graphs of bipartite graphs ([1]),
and for line graphs of order n and size (2+ ε)n of a bipartite graph, with
ε > 0 fixed ([1]).
STABLE CUTSET*(n, m): Given a graph G with n vertices and m
edges. Does G contain a stable cutset?
Problem 27.2 Is STABLE CUTSET*(n, m) in P or NP-complete,
for m ∈ {2n − 3, 2n − 2, 2n − 1, 2n}?
The following problem is motivated by conjecture 26.1:
Problem 27.3 Let G be a graph with n vertices and at most 3n− 7
edges. Does there exist a cutset C of G with at most |C| − 1 edges?
References:
[1] V. B. Le, B. Randerath, On stable cutsets in line graphsLecture Notes in Computer Science 2204 (2001), 263-271
[2] G. Chen, X. Yu, A note on fragile graphsDiscrete Math. 249 No. 1-3 (2002), 41-43
23
DIGRAPHS & HYPERGRAPHS
28 Difference Labelling of Digraphs
(Martin Sonntag)
A digraph G is a difference digraph iff there exists an S ⊂ IN+ such
that G is isomorphic to the digraph DD(S) = (V, E), where V = S and
E = {(i, j) : i, j ∈ V ∧ i − j ∈ V }.For some classes of digraphs, e.g. certain trees, oriented cycles, tour-
naments etc., it is known, under which conditions these digraphs are
difference digraphs (cf. [1], [2]).
In the undirected case the composition of difference labellings of cycles
with prickles (i.e. hanging edges) and caterpillars (i.e. paths with
prickles) to cacti makes no problems (cf. [3]), but in the directed case only
partial results seem to be possible. Even to add prickles to a cycle causes
a lot of problems and can result in a difference digraph or not (many
cases must be considered, e.g. whether or not there are ingoing/outgoing
edges at adjacent vertices of the cycle, where the direction of the edges
along the cycle is important, too).
Problem 28.1 Find classes of oriented cacti which are difference
digraphs.
References:
[1] R.B. Eggleton, S.V. Gervacio, Some properties of difference graphsArs Combinatoria 19A (1985), 113-128
[2] M. Sonntag, Difference labelling of the generalized source-join of digraphsTU Bergakademie Freiberg, Faculty of Mathematics and Computer Science,Preprint 2003-03 (2003), 1-18
[3] M. Sonntag, Difference labelling of cactiDiscussiones Mathematicae Graph Theory, to appear in Vol. 23 (2003)
24
DIGRAPHS & HYPERGRAPHS
29 Sum Hypergraphs
(Hanns Teichert)
Let S ⊆ IN+ be finite. The hypergraph Hd(S) has vertex set S and edge
set
{e ⊆ S | |e| = d ∧∑
v∈e
v ∈ S}. A d-uniform hypergraph H is a sum
hypergraph iff there is an S such that H ∼= Hd(S). It is known that
the edge set of the sum graph H2({1, . . . , n}) has maximum cardinality
emax(n, 2) possible in the class of sum graphs with n vertices. This is not
true for Hd({1, . . . , n}) in case of d ≥ 3.
Problem 29.1 Determine the number emax(n, d) for d ≥ 3. Find
a set S ′ that induces a d-uniform sum hypergraph Hd(S ′) having n
vertices and an edge set with maximum cardinality emax(n, d).
25