9th C5 GRAPH THEORY WORKSHOP

”Cycles, Colorings, Cliques, Claws and Closures”

Kurort Rathen, 9.-13.05.2005

SELECTED PROBLEMS

http://www.mathe.tu-freiberg.de/

math/inst/theomath/WorkshopRathen.html

COLORINGS

1 Vertex Colouring and Forbidden Subgraphs

(Ingo Schiermeyer, Bert Randerath)

It is not difficult to colour the vertices of a graph in polynomial time

using at most ∆(G) + 1 colours, where ∆(G) denotes the maximum

vertex degree of a given graph G. Moreover, the classical theorem of

Brooks states that χ(G) ≤ ∆(G) unless G is a complete graph or an

odd cycle.

Reed [5] (see also [3]) believes that this result is just the tip of iceberg.

He conjectured that the chromatic number is bounded by the average of

the trivial upper and lower bound.

Conjecture 1.1 For any graph G of maximum degree ∆,

χ(G) ≤⌈

∆ + 1 + ω

2

⌉

The Chvatal graph [2], the smallest 4-regular, triangle-free graph of or-

der 12 with chromatic number 4, shows that the rounding up in this

conjecture is necessary.

In [5] it was observed that Conjecture 1.1 is also valid for all graphs

G = (V, E) with ∆(G) = |V | − 1. The main result in [5] asserts that

if ∆ is sufficiently large and ω is sufficiently close to ∆, then Conjecture

1.1 holds.

Theorem 1.1 ([5]) There is a constant ∆0 such that for ∆ ≥ ∆0

and if G is a graph of maximum degree ∆ having clique number ω

with ω ≥ b(1− 170000000)∆c, then χ(G) ≤ ∆+1+ω

2 .

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A related result due to Reed [6] thereby mainly proving a conjecture of

Beutelsbacher and Hering [1] asserts that for sufficiently large ∆, any

graph with maximum degree at most ∆ and no cliques of size ∆ has a

∆− 1 colouring.

Using Ramsey numbers, Conjecture 1.1 has been recently verified as fol-

lows.

Theorem 1.2 ([4]) For every k ≥ 3 there is a constant ck such that

if G is a graph of order n with clique number ω and maximum degree

∆ ≥ 2nk + ckω

k−1, then χ(G) ≤ ∆+1+ω2 .

Remark: For every k a vertex colouring with at most ∆+1+ω2 colours

can be found in time O(nk+1) by an algorithm implicitly given by the

proof of Theorem 1.2.

By a similar approach we are able to show that Conjecture 1.1 holds for

all graphs G with maximum degree ∆(G) = n − k if the independence

number of G satisfies α(G) ≥ k + 1.

Theorem 1.3 ([4]) Let G be a graph with maximum degree ∆ =

n−k for some k ≥ 1, independence number α and clique number ω.

If G satisfies α ≥ k + 1, then χ(G) ≤ ∆+1+ω2 .

References:

[1] A. Beutelsbacher and P.R. Hering, Minimal graphs for which the chromatic number equals the maximaldegreeArs Combin. 18 (1984), 201-216

[2] V. Chvatal, The smallest triangle-free, 4−chromatic, 4−regular graphJ. Combin. Theory 9 (1970), 93-94

[3] M. Molloy and B.A. Reed, eds., Graph Colourings and the Probabilistic MethodAlgorithms and Combinatorics 23, Springer-Verlag Berlin (2002)

[4] B. Randerath and I. Schiermeyer, Algorithmic Bounds for the Chromatic NumberPreprint 2004.

[5] B.A. Reed, ω, ∆ and χJ. Graph Theory 27 No.4 (1998), 177-212

[6] B.A. Reed, A Strengthening of Brooks’ TheoremJ. Combin. Theory Ser. B 76 No.2 (1999), 136-149

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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],[2])

3. 3− COL(P7−free) can be decided in polynomial time for K3−free

and flower-free graphs ([3]), where a flower is an induced C5 with an

attached path of length 2.

References:

[1] B. Randerath, I. Schiermeyer and M. Tewes, Three-colourability and forbidden subgraphs II: PolynomialalgorithmsDiscrete Math. 251 No.1-3, (2002), 137-153

[2] B. Randerath and I. Schiermeyer, 3-Colorability ∈ P for P6-free GraphsDiscrete Applied Mathematics 136 (2004), 299-313

3 J. Bottcher, 3-Farben von P7, ∆, flower-freien Graphen in polynomieller ZeitStudienarbeit, Humboldt-Universitat zu Berlin, August 2003

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3 Fruit Salad

(Andras Gyarfas)

Conjecture 3.1 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

4

COLORINGS

4 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 4.1 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. 14 No.1 (2001), 121-137

[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

5

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5 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 + 12k+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 5.1 Determine classes of graphs G aside from cycles such

that

χ′′c (G) < χ′′(G).

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6 List Colorings of Integer Distance Graphs

(Arnfried Kemnitz)

Let D be a subset of the positive integers N. The integer distance graph

G(Z, D) = G(D) is defined as the graph with the set of integers as vertex

set, V (G(D)) = Z, 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 6.1 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 22 (2002), 149-158

[2] M. Voigt, Colouring of distance graphsArs Combinatoria 52 (1999), 3-12

[3] X. Zhu, Distance graphs on the real linemanuscript, 1996

7

COLORINGS

7 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 7.1 Does there is an absolute constant c such that

ch(G×H) ≤ max{ch(G), ch(H)} + c?

Question 7.2 If the answer is YES, then how big is c? Is c = 1?

8

COLORINGS

8 Neighbours distinguishing indexof plane bipartite graphs

(Keith Edwards, Mirko Hornak, Mariusz Wozniak)

Let G be a finite simple graph with no component K2. Let C be a finite

set of colours and let ϕ : E(G) → C be a proper edge colouring of

G. The colour set of a vertex v ∈ V (G) with respect to ϕ, in symbols

Sϕ(v), is the set of colours of edges incident with v. The colouring ϕ is

neighbours distinguishing if Sϕ(x) 6= Sϕ(y) for any xy ∈ E(G). The

neighbours distinguishing index of the graph G, in symbols ndi(G), is

the smallest number of colours in a neighbours distinguishing colouring

of G. Neighbours distinguishing index has been introduced in [2], where

the authors have conjectured that ndi(G) ≤ ∆(G) + 2 for any simple

connected graph G nonisomorphic to C5 on at least three vertices. (It is

easy to see that ndi(C5) = 5.) Their conjecture has been confirmed in

[1] for graphs of maximum degree 3 and for bipartite graphs. We have

proved that ndi(G) ≤ ∆(G) + 1 for any plane bipartite graph G with

∆(G) ≥ 12.

Problem 8.1 Find the minimum integer ∆ ≥ 4 such that ndi(G) ≤∆(G) + 1 for any plane bipartite graph G with ∆(G) ≥ ∆.

References:

[1] P. N. Balister, E. Gyori, J. Lehel and R. H. Schelp, Adjacent vertex distinguishing edge-coloringsmanuscript (2002)

[2] Z. Zhang, L. Liu, and J. Wang, Adjacent strong edge coloring of graphAppl. Math. Lett. 15 (2002), 623-626

9

COLORINGS

9 General neighbour-distinguishing indexof a graph

(Mirko Hornak, Mariusz Wozniak)

Let G be a finite simple graph with no component K2. Let k ∈ Z,

k ≥ 2, and let ϕ : E(G) → {1, . . . , k} be an edge colouring of G.

The terms colour set and neighbour-distinguishing are defined as in

the previous problem. The general neighbour-distinguishing index of

the graph G, in symbols gndi(G), is the smallest number of colours in a

neighbour-distinguishing edge colouring of G.

The general neighbour-distinguishing index is a relaxation of two known

graph invariants. If Sϕ(x) 6= Sϕ(y) is required for any two distinct

vertices x, y, the corresponding parameter χ0(G), called the point-

distinguishing chromatic index of G, has been introduced by Harary

and Plantholt in [1]. On the other hand, if only proper neighbour-

distinguishing colourings are considered, the neighbour-distinguishing

index of G, symbolically ndi(G), is obtained. The investigation of this

invariant has been started by Zhang et al. in [3].

We have proved in [2] that gndi(G) ≤ 3 if G is bipartite, gndi(Pn) =

2 (for the n-vertex path Pn) iff n ≡ 1 ( mod 2), gndi(Cn) ≤ 3 and

gndi(Cn) = 2 iff n ≡ 0 ( mod 4). Our main result states that gndi(G)

is bounded from above by χ(G) + 2 provided that χ(G) ≥ 3.

Conjecture 9.1 If G is a connected graph with χ(G) ≥ 3, then

gndi(G) ≤ χ(G) + 1.

References:

[1] F. Harary and M. Plantholt, The point-distinguishing chromatic indexin: F. Harary, J.S. Maybee (Eds.), Graphs and Applications, Wiley-Inter-science, New York (1985) , 147-162

[2] M. Hornak and M. Wozniak, General neighbour-distinguishing index of a graphmanuscript, March 2005, available as IM Preprint A3-2005, http://umv.science.upjs.sk/preprints/index.htmland as Preprint MD 011 (2005), http://www.ii.uj.edu.pl/preMD/index.htm

[3] Z. Zhang, L. Liu, and J. Wang, Adjacent strong edge coloring of graphAppl. Math. Lett. 15 (2002), 623-626

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CYCLES

10 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.

Problem 10.1 Determine the maximum number of cycle lengths

that can be missing in a claw-free graph on n vertices with complete

closure.

Conjecture 10.1 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.

Recent development (2005): counterexamples to conjecture 18.1 have

been found; however, all these counterexamples have connectivity κ ≤5. The conjecture remains open (and probably true) for graphs with

connectivity κ ≥ 5.

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CYCLES

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 Math. 236 (2001), 325-338

11 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 11.1 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

12

CYCLES

12 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 12.1 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.

13 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 13.1

f (n, k, k) =2

kn + ck ?

13

CYCLES

Known:

1. f (n, k, 1) = 2

(k2)

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

14 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 14.1 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 verticesDiscrete Math. 286 (2004), 95-98

[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

14

CYCLES

15 Cycles containing k−connected VertexSubsets and Independent Edge Subsets

(Jochen Harant)

Given k ∈ N, a graph G, and X ⊆ V (G). X is k-connected in G

if G − S has a component containing X for every S ⊆ V (G) with

|S| ≤ k − 1.

Remark: Even in a planar graph this ’regional’ connectedness of a

vertex set X can be arbitrarily large (in opposition to the usual global

connectedness of a planar graph, i.e. if X = V (G))!

Theorem 15.1 Let G be a planar graph, X ⊆ V (G), such that X

is 4-connected in G, and E ⊂ E(G[X ]), such that |E| ≤ 2 and E is

independent if |E| = 2.

Then there is a cycle of G containing X ∪ E.

Theorem 15.2 Let G be a graph, X ⊆ V (G), such that |X| ≥ 7

and X is (|X| − 3)-connected in G, and E ⊂ E(G[X ]), such that

|E| = 3 and E is independent.

Then there is a cycle of G containing X ∪ E.

(Not true if X is only (|X| − 4)-connected in G.)

Problem 15.1 Is there a positive constant c such that the following

holds ?

A planar graph G with X ⊆ V (G), X is c-connected in G, E ⊂E(G[X ]), |E| = 3, and E is independent, has a cycle containing

X ∪ E.

15

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16 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 16.1 (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 16.1 (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.

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17 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 17.1 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

17

INDEPENDENT SETS & INTERSECTIONS

18 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 18.1 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.

19 Nonempty Intersections

(Stanislav Jendrol’, ZdzisÃlaw Skupien)

Conjecture 19.1 In a connected (2−connected) graph G any three

longest paths (cycles) have a nonempty intersection.

Known: In a connected (2−connected) graph any k longest paths (cy-

cles) for each k ≥ 7 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

[2] Z. Skupien, Smallest sets of longest paths with empty intersectionCombin. Probab. Comput. 5 (1996), 429-436

18

DECOMPOSITIONS & PARTITIONS

20 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 orders

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 (see [2]).

Conjecture 20.1 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.

Conjecture 20.1 has been proved in [1].

References:

[1] D. Barth and H. Fournier, A degree bound on decomposable treesRapports de recherche du Laboratoire PRISM, Rapport #2004/59, June 2004.

[2] M. Hornak and M. Wozniak, Arbitrarily vertex decomposable trees are of maximum degree at most sixOpuscula Math. 23 (2003), 49-62

19

DECOMPOSITIONS & PARTITIONS

21 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 21.1 [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 21.1 Does the PPC hold for τ (G) = n− 4?

Known: asymptotic result (Frick, Schiermeyer, [1])

For a graph G with τ (G) = n − p for some fixed p ≥ 4 the PPC holds

provided that n ≥ p(10p− 3).

References:

[1] M. Frick and I. Schiermeyer, An asymptotic Result for the Path Partition ConjecturePreprint 2003, University of South Africa.

20

DECOMPOSITIONS & PARTITIONS

22 Which graph invariants are ARHP?

(Peter Mihok, Gabriel Semanisin)

Let ϕ(G) be a graph invariant. A graph G is called ϕ-partitionable if,

for any pair of positive integers (k1, k2) satisfying k1 + k2 ≥ ϕ(G) − 1,

there exists a partition {V1, V2} of V (G) such that ϕ(G[V1]) ≤ k1 and

ϕ(G[V2]) ≤ k2.

The next well-known results for the maximum degree ∆(G) provides an

illustration of such an invariant:

Theorem 22.1 (Lovasz, 1966) Every graph G is ∆-partitionable.

The following problem have been formulated by I.Schiermeyer during the

workshop Hereditarnia 2003 (see [1]):

Problem 22.1 Which other partition concepts/problems of this type

do exist?

We have investigated a problem related to the previous one: Let P be a

hereditary property of graphs. Given a graph invariant ϕ, we define the

associated invariant of the property P in the following manner:

ϕ(P) = min{ϕ(F ) : F ∈ F(P)}.The motivation for the investigation of invariants related to hereditary

graph properties comes from extremal and chromatic graph theory. The

classical Erdos-Stone-Simonovits formula provides a relationship between

the maximum number of edges in a P-maximal graph of order n and the

invariant χ(P) - the chromatic number of P (see e.g. [7]).

LetP1,P2, . . . ,Pn be any properties of graphs. A vertex (P1,P2, . . . ,Pn)-

partition of a graph G is a partition (V1, V2, . . . , Vn) of V (G) such that

for each i = 1, 2, . . . , n the induced subgraph G[Vi] has the property Pi.

The propertyR = P1◦P2◦ . . . ◦Pn is defined as the set of all graphs having

a vertex (P1,P2, . . . ,Pn)-partition. If a property R can be expressed as

the product of at least two properties, then it is said to be reducible;

otherwise it is called irreducible (for more details see e.g. [3]).

21

DECOMPOSITIONS & PARTITIONS

A. Berger [2] proved that any reducible additive hereditary property of

graphs has infinitely many minimal forbidden graphs. But only very

little is known about the structure of F(P ◦Q), even in the case when the

structure of F(P) and F(Q) is known. Moreover, A. Farrugia proved

in [4], that recognizing whether a graph belongs to a property P ◦Q (i.e.

recognizing whether it contains a graph from F(P ◦Q) as a subgraph) is

polynomial only in the simplest case : if the property P ◦Q is the property

“to be bipartite”. Useful information on the structure of F(P ◦Q) can be

obtained by investigation of graph invariants associated with the property

P ◦Q.

We say that a graph invariant ϕ is additive with respect to reducible

hereditary properties (abbreviated by ARHP) if for any reducible prop-

erty P ◦Q the equality ϕ(P ◦Q) = ϕ(P) + ϕ(Q) is valid.

In [5] we proved that the subchromatic number ψ(G) = χ(G) − 1 is

ARHP and we stated the following problem:

Problem 22.2 Which graph invariants are ARHP?

In [6] we presented a necessary and sufficient condition for a graph invari-

ant to be ARHP and we proved that amongst the others the degeneracy

number and tree-width are ARHP. Our investigation stimulates us to

formulate the following more specific problem:

Problem 22.3 Is the choice number ch(G) ARHP?

References:

[1] http://www.science.upjs.sk/hereditarnia

[2] A.J. Berger, Minimal forbidden subgraphs of reducible graph propertiesDiscuss. Math., Graph Theory 21 (2001) 111–117.

[3] M. Borowiecki, I. Broere, M. Frick, P. Mihok and G. Semanisin, Survey of hereditary properties of graphs,Discussiones Mathematicae - Graph Theory 17 (1997) 5–50.

[4] A. Farrugia, Vertex-partitioning into fixed additive induced-hereditary properties is NP-hardElectron. J. Combin. 11 (2004), R46, 9 pp.

[5] P. Mihok and G. Semanisin, On the chromatic number of reducible hereditary propertiesStud. Univ. Zilina Math. Ser. 16 (2003), 51-54.

[6] P. Mihok and G. Semanisin, On invariants of hereditary graph properties(to appear in a special volume of Discrete Mathematics devoted to the workshop Cycles & Colourings 2003).

[7] M. Simonovits, Extremal graph theory, In L.W. Beineke and R.J. Wilson, editors, Selected topics in graphtheory, volume 2 (Academic Press London, 1983) 161–200.

22

FACTORS & CUTSETS

23 2-Factor with k Components

(Ralph Faudree, Ronald Gould, Mike Jacobson)

(Ingo Schiermeyer)

Theorem 23.1 (Dirac)

If δ(G) ≥ n2 then G is hamiltonian (n = |V (G)|).

Question 23.1 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 componentsK5 − eK5 − e

Conjecture 23.1 (Gould) δ(G) ≥ const · log n suffices

For a graph G, we denote by α(G) and κ(G) the independence number

and the connectivity of G, respectively. If the inequality α(G) ≤ κ(G)

holds, we say that G satisfies the Chvatal-Erdos condition. It is a well-

known fact that a 2-connected graph satisfying the Chvatal-Erdos con-

dition has a hamiltonian cycle.

Theorem 23.2 (Chvatal– Erdos Theorem) Every 2-connected

graph G satisfying α(G) ≤ κ(G) has a hamiltonian cycle.

Conjecture 23.2 (Kaneko and Yoshimoto, [1] ) For each pos-

itive integer k, there exists an integer n such that every 2-connected

graph G of order at least n with α(G) ≤ κ(G) has a 2-factor with k

components.

23

FACTORS & CUTSETS

Recently, Chen, Gould, Karawabayashi, Ota, Saito and Schiermeyer par-

tially answered the question posed by Kaneko and Yoshimoto.

Theorem 23.3 Let G be a 2-connected graph of order n satisfying

α(G) = a ≤ κ(G). Then (1) if n ≥ k · r(a + 4, a + 1), then G has

a 2-factor with k components, where r(m,n) is the Ramsey number,

and (2) if n ≥ r(2a + 3, a + 1) + 3(k− 1), then G has a 2-factor with

k components such that all components but one have order three.

References:

[1] A.Kaneko and K.Yoshimoto, A 2-factor with two components of a graph satisfying the the Chvatal-Erdoscondition, J. Graph Theory 43 (2003), 269–279.

[2] G.Chen, R.J.Gould, K.Karawabayashi, K.Ota, A.Saito and I.Schiermeyer, The Chvatal-Erdos condition and2-Factors with a Specified Number of Components, preprint 2005.

24 Acyclic Cutsets

(Atsushi Kaneko)

Conjecture 24.1 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

24

FACTORS & CUTSETS

25 Stable Cutsets

(Bert Randerath)

STABLE CUTSET: Given a graph G. Does G contain a stable cutset?

Problem 25.1 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 25.2 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 24.1:

Problem 25.3 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

25

DIGRAPHS & HYPERGRAPHS

26 Difference Labelling of Digraphs

(Martin Sonntag)

A digraph G is a difference digraph iff there exists an S ⊂ N+ 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 26.1 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 24(3) (2004), 509-527

26

DIGRAPHS & HYPERGRAPHS

27 Sum Hypergraphs

(Hanns Teichert)

Let S ⊆ N+ 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 27.1 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).

27

DIGRAPHS & HYPERGRAPHS

28 Domination hypergraphs of tournaments

(Martin Sonntag, Hanns Teichert)

Let D = (V,A) be a digraph. A subset V ′ ⊆ V is called a dominating

set iff ∀x ∈ V \V ′ ∃ y ∈ V ′ : (y, x) ∈ A. The domination graph D(D)

has vertex set V and its edges are the dominating sets of cardinality

two (see for instance [1]). As a natural generalization the domination

hypergraph DH(D) also has vertex set V and its edges are all minimal

dominating sets V ′ with |V ′| ≥ 1.

There are many interesting results on domination graphs of tournaments

Tn, e.g. in general D(Tn) is not connected (see for instance [2]).

Conjecture 28.1 The domination hypergraph DH(Tn) of a tourna-

ment Tn is always connected.

The conjecture is true for n ≤ 6. We tested hundreds of bigger examples

(up to n = 23) by Mathematica routines and found no counterexample.

It is easy to prove that every nontrivial component of DH(Tn) contains

at least three edges. A first step to verify the conjecture could be the

investigation of regular tournaments.

References:

[1] Fisher, Lundgren, Merz, Reid, The domination and competition graphs of a tournamentJ. Graph Theory 29 (1998), 103-110

[2] Fisher, Lundgren, Guichard, Reid, Domination graph of tournaments with isolated verticesArs Combinatoria 66 (2003), 299-311

28

WEIGHT OF GRAPHS

29 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 29.1 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 [2] where the reader can find more information 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

29

GRAPH EDITING

30 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 30.1 What if f (n,H) for H is a specific class of graphs,

for instance with fixed chromatic number?

Question 30.2 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 (cf. [1]).

References:

[1] M. Axenovich, A. Kezdy, R. Martin, Editing distance of graphspreprint 2003.

30

NOTES

31