16th C5 Graph Theory Workshop"Cycles, Colourings, Cliques, Claws and Closures"

SELECTED PROBLEMS

http://www.mathe.tu-freiberg.de/inst/theomath/rindex.php?seite=rathen

Colourings

1 Minimum order of k−chromatic

Kr+1−free graphs

(Anja Kohl)Let nr(k) denote the smallest possible number of vertices of a graph G with chro-matic number χ(G) = k and clique number ω(G) 6 r. Obviously, nr(k) = k forr > k. So we are only interested in the case r 6 k− 1.

It is known that n2(k) has order of magnitude k2 · logk.Question 1.1. What order of magnitude has nr(k) for r > 3?For k 6 3 + r all exact values for nr(k) are known, in particular: n2(4) = 11 [1],n2(5) = 22, n3(5) = 11 [2] and n3(6) = 16 [4].Problem 1.1. Determine n2(6), n3(7), and n4(8).(known: 30 6 n2(6) 6 45, 20 6 n3(7) 6 31, 17 6 n4(8) 6 22)

Known ([3]): For k sufficient large and r > k− 11, we have the following results:• nk−i(k) = k+ 2i for k > 3i, i = 1, 2

• nk−i(k) = k+ 2i for k > 2i+ 1, i = 3, 4, 5, 6

• nk−i(k) = k+ 2i− 1 for k > 2i, i = 7, 8

• nk−i(k) = k+ 2i− 2 for k > 2i− 1, i = 9

• nk−i(k) = k+ 2i− 3 for k > 2i− 2, i = 10, 11

• nk−i(k) = k+ 2i− 4 for k > 2i− 3, i = 13.Problem 1.2. Determine nk−12(k) for k > 22.(known for k > 22: k+ 20 6 nk−12(k) 6 k+ 21).The value of nk−12(k) for k > 22 depends on the Ramsey number R(10, 3). It isknown that 40 6 R(10, 3) 6 43. If R(10, 3) ∈ {42, 43}, then nk−12(k) = k+ 20 fork > 22, otherwise nk−12(k) = k+ 21.

References:[1] V. Chvatal, The minimality of the mycielski graph

Graphs and Combinatorics, Springer-Verlag, Berlin and New York, (1974), 243-246.

[2] T. R. Jensen, G. F. Royle, Small graphs with chromatic number 5: a computer searchJ. Graph Theory 19 (1995), 107-116.

[3] A. Kohl, On k−chromatic Kr+1−free graphs: I. Determining the minimum ordermanuscript (2010), submitted to Journal of Graph Theory.

[4] private communication to S. Radziszowski.

2

Colourings

2 Chromatic number of graphs with

two odd cycle lengths

(Stephan Matos Camacho)Let Co(G) denote the set of odd cycle lengths contained by a graph G. Gyarfas([2]) proved that G is colourable with at most 2k+ 1 colours if |Co(G)| = k, unlessG contains a K2k+2. In [1] we observed

Theorem 2.1. If G only contains two consecutive odd lengths greater than 3, sayCo(G) = {2m+ i : i ∈ {−1, 1},m > 3}, then G is 4-colourable.

Question 2.1. Let G be a 2-connected graph with Co(G) = {2m + i : i ∈{−1, 1},m > 3} and minimal degree at least 3. Is χ(G) < 4?

In [5] a first part of the question could be answered:

Theorem 2.2. IfG contains only odd cycles of length 5 and 7, thenG is 3-colourable.

The proof is based on known results on the structure of k-critical graphs due toDirac and Toft. The rest of the questions still remains open.

Question 2.2. LetG be a 2−connected graph with Co(G) = {2m+1, 2n+1 : m >

n+ 1,n > 2} and minimal degree at least 3. Is χ(G) < 5?

References:[1] S. Matos Camacho, Colourings of graph with prescribed cycle lengths

diploma thesis (2006).

[2] A. Gyarfas, Graphs with k odd cycle lengthsDiscrete Math. 103 (1992), 41-48.

[3] S. Matos Camacho, I. Schiermeyer, Colourings of graphs with two consecutive odd cycle lengthsDisc. Math. 309 (15), 4916–4919.

[4] H.-J. Voss, Cycles and Bridges in GraphsDeutscher Verlag der Wissenschaften, Berlin (1991).

[5] T. Kaiser, O. Rucky and R. Skrekovski, Graphs with odd cycle lengths 5 and 7 are 3-colorablemanuscript (2008).

3

Colourings

3 b-colourings of graphs

(Mais Alkhateeb, Anja Kohl)Let G be a simple and undirected graph of order n. A b-coloring of G is a propervertex coloring such that there is a vertex in each color class, which is adjacentto at least one vertex in every other color class. Such a vertex is called a color-dominating vertex. The b-chromatic number of a graph G, denoted by χb(G), isthe largest k such that there is a b-coloring of G by k colors.An interesting problem is to characterize the d-regular graphsG satisfying χb(G) =d + 1. This equality is e.g. satisfied for every d-regular graph with at least 2d3

vertices (Cabello and Jakovac ([3])) and for every d-regular graph with girth g > 6(independently proved by El Sahili et al., Kouider et al., and Blidia et al.).

Conjecture 3.1. (Blidia, Maffray, and Zemir ([2])) Every d-regular graph G with girthg > 5 which is not the Petersen graph satisfies χb(G) = d+ 1.

A graph G is called b-continuous if it has a b-coloring by k colors for all integers ksatisfying χ(G) 6 k 6 χb(G). There exist graphs which are not b-continuous, e.g.every (r− 1)-factor of the complete bipartite graph Kr,r for r > 4.Recently we could verify that graphs G with minimum degree δ(G) > n − 3 areb-continuous ([1]). Moreover, there exist non-b-continuous graphs G of order nand minimum degree δ(G) = n− 5. So we ask:

Question 3.1. Is every graph G with minimum degree δ(G) = n−4 b-continuous?

Since all non-b-continuous graphs that are known so far contain a claw as aninduced subgraph, we ask:

Question 3.2. Does there exist a claw-free graph that is not b-continuous?

We conjecture that there is no such graph. This is reason to pose the followingconjecture:

Conjecture 3.2. Line graphs are b-continuous.

So far we proved it for line graphs of 2-degenerate graphs.

References:[1] M. Alkhateeb, A. Kohl, "Upper bounds on the b-chromatic number and results for restricted graph classes"

manuscript (2010), accepted by Discussiones Mathematicae Graph Theory.

[2] M. Blidia, F. Maffray, Z. Zemir, "On b-colorings in regular graphs"Discrete Appl. Math. 157 (2009), 1787-1793.

[3] S. Cabello, M. Jakovac, "On the b-chromatic number of regular graphs", preprint (2010).

4

Colourings

4 On the cyclic chromatic number of

3-connected plane graphs

(Mirko Hornak)The cyclic chromatic number of a plane graph G, in symbol χc(G), is a minimumnumber of colours in such a vertex colouring ofG that distinct vertices incident witha common face receive distinct colours. If G is 2-connected, then χc(G) > ∆∗(G),where ∆∗(G) is the maximum face degree of G.On the other hand, no 3-connected plane graphG is known withχc(G) > ∆

∗(G)+2.Plummer and Toft ([7]) proved that χc(G) 6 ∆∗(G) + 9 and conjectured (PTC) thatχc(G) 6 ∆∗(G) + 2 for any 3-connected plane graph G.Let PTC(d) denote PTC restricted to 3-connected plane graphs G with ∆∗(G) = d.It is known that PTC(d) is true for d = 3 (Four Colour Theorem), d = 4 (Borodin[1]), d ∈ {18, . . . , 23} (Hornak and Zlamalova [6]) and d > 24 (Hornak and Jendrol’[5]). For ∆∗(G) > 60 Enomoto et al. ([4]) obtained the best possible inequality:χc(G) 6 ∆∗(G) + 1 (graphs of pyramids show that the bound ∆∗(G) + 1 cannot beimproved).The best general upper bound known so far is due to Enomoto and Hornak ([3]),namely χc(G) 6 ∆∗(G) + 5.

Problem 4.1. Prove PTC(d) for some d ∈ {5, . . . , 17}.

References:[1] O. V. Borodin, Solution of Ringel’s problem on vertex-face coloring of plane graphs and coloring of 1-planar graphs

(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 graphsJ. Graph Theory 62 (2009), 1-25.

[4] H. Enomoto, Hornak and S. Jendrol’, Cyclic chromatic number of 3-connected plane graphsSIAM J. Discrete Math. 14 (2001), 121-137.

[5] M. Hornak and S. Jendrol’, On a conjecture by Plummer and ToftJ. Graph Theory 30 (1999), 177-189.

[6] M. Hornak, J. Zlamalova, Another step towards proving a conjecture by Plummer and ToftDiscrete Math. 310 (2010), 442-452.

[7] M. D. Plummer and B. Toft, Cyclic coloration of 3-polytopesJ. Graph Theory 11 (1987), 507-515.

5

Colourings

5 The circular total chromatic number

(Andrea Hackmann, Arnfried Kemnitz)A k−total colouring of a simple graph G is an assignment of k colours to the ver-tices and edges of G such that the neighbored elements - two adjacent vertices ortwo adjacent edges or a vertex incident to an edge - are coloured differently. Theminimum number k for which a graph G admits a k−total colouring is the totalchromatic number χ ′′(G) of G.

If k and d are positive integers with k > 2d then a (k,d)−total colouring of agraph G is an assignment c of colours {0, 1, . . . , k − 1} to the vertices and edgesof G such that d 6 |c(xi) − c(xj)| 6 k − d whenever two elements xi and xj areneighbored. The circular total chromatic number χ ′′c (G) of G is defined as theinfimum of fractions k

dfor all (k,d)−total colourings of G:

χ ′′c (G) = inf{k

d: G has a (k,d) − total colouring.

}

Obviously, a (k, 1)−total colouring is a k−total colouring of G which implies thatχ ′′c (G) 6 χ

′′(G). For cyclesCp 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 multipartitegraphs the total chromatic number and the circular total chromatic number coin-cide.

Problem 5.1. Determine classes of graphs G aside from cycles such that

χ ′′c (G) < χ′′(G).

6

Colourings

6 Choice number of cartesian products

(Mieczysław Borowiecki, Stanislav Jendrol’)Let ch(G) denote the choice number of G and let G×H be the Cartesian Productof graphs G and H.Galvin (1995) proved that ch(Kn×Kn) = n and solved in this way the old Dinitz’sconjecture. (See Diestel’s book where the solution is presented as a result on theedge list colouring of bipartite multigraphs.)

Let G and H be graphs. Clearly,

max{ch(G), ch(H)} 6 ch(G×H).

Question 6.1. Does there is an absolute constant c such that

ch(G×H) 6 max{ch(G), ch(H)}+ c?

Question 6.2. If the answer is YES, then how big is c? Is c = 1?

Comment: In [1] we have shown that the above conjecture is not true in a generalcase.However, we suspect that the following two conjectures hold:

Conjecture 6.1. There exists a constant A such that the following holds for everypair of graphs G and H:

ch(G×H) 6 A(ch(G) + ch(H)).

Conjecture 6.2. LetG andH be two graphs with maximum degree at most ∆. Then

ch(G×H) 6 ∆+ o(∆).

References:[1] M. Borowiecki, S. Jendrol’, D. Kral and J. Miskuf, List Coloring of Cartesian Products of Graphs

Discrete Mathematics 306 (2006), 1955-1958.

7

Colourings

7 Colouring vertices of plane graphs under

restrictions given by faces

(Stanislav Jendrol’)Consider a vertex colouring of a connected plane graph G. A colour c is used ktimes by a face α of G if it appears k times along the facial walk of α. Two naturalproblems arise.

1. A vertex colouring ϕ is a weak parity vertex colouring of a connected planegraph G with respect to its faces if each face of G uses at least one colour an oddnumber of times. Problem is to determine the minimum number χw(G) of coloursused in a wpv colouring of G.

In [1] it is proved that χw(G) 6 4 for every connected plane graphGwith minimumdegree at least 3. We strongly believe that the following holds.

Conjecture 7.1. Let G be a connected plane graph of minimum face degree at least3. Then

χw(G) 6 3.

The Conjecture is true for 2-connected cubic plane graphs, see [1].

2. A vertex colouring ϕ is a strong parity vertex colouring of a 2-connected planegraph G with respect to the faces of G if each face of G that uses a colour then ituses an odd number of times. Problem is to find the minimum number χs(G) ofcolours used in an spv colouring of G. We believe that

Conjecture 7.2. There is a constant k such that for every 2-connected plane graphG

χs(G) 6 k.

We do not know any 2-connected plane graph H with χs(H) > 7. Hence, webelieve that k = 6 in the above conjecture.

References:[1] J. Czap, S. Jendrol’, Colouring vertices of plane graphs under restrictions given by faces, Discussiones Math.

Graph Theory (submitted).

8

Colourings

8 List colourings of integer distance graphs

(Arnfried Kemnitz)LetD be a subset of the positive integersN. 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 theso-called distance set D.

General bounds for the chromatic number of integer distance graphs are

2 6 χ(G(D)) 6 |D|+ 1.

Voigt ([4]) and Zhu ([5]) determined χ(G(D)) if |D| = 3 :

If D = {x,y, z} consists of integers whose greatest common divisor equals 1, thenχ(D) = 4 if and only if D = {1, 2, 3n} or D = {x,y, x + y} and x . y (mod 3). Ifx,y, z are odd then χ(D) = 2. For all other 3−element distance sets D it holdsχ(D) = 3.

General bounds for the list chromatic number (choice number) of integer distancegraphs are χ(D) 6 ch(D) 6 |D|+ 1 (Kemnitz, Marangio 2001).

Question 8.1. Does there exist a 3−element distance set such that ch(D) < 4?

References:[1] A. Hackmann, A. Kemnitz, Circular total colorings of graphs

Congr. Numer. 158 (2002), 43-50.

[2] A. Hackmann, A. Kemnitz, Circular total colorings of cubic circulant graphsJ. Combin. Math. Combin. Comput. 49 (2004), 65-72.

[3] A. Kemnitz, M. Marangio, Edge colorings and total colorings of integer distance graphsDiscussiones Mathematicae Graph Theory 22 (2002), 149-158.

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

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

9

Colourings

9 Improper colouring of planar graphs without

4- and 5-cycles

(Andre Raspaud)Definition 9.1. A simple graphs G = (V ,E) is called (d1,d2, . . . ,dk)-colourable ifthere is a partition V = V1 ∪ V1 ∪ · · · ∪ Vk of the vertex set such that the maximumdegree of Gi = [Vi] is at most di for i = 1, . . . ,k.

Conjecture 9.1 (Steinberg, 1976). Every planar graph without 4- and 5-cycles is3-colourable ((0,0,0)-colourable)

Question 9.1. Let G be a planar graph without 4- and 5-cycles. Can we prove(1, 0, 0)- or (1, 1, 0)- or (2, 0, 0)- or . . . -colourability of G?

Remark: (7, 0, 0)-colourability seems ok.

The (1, 1, 1)-colourability of such graphs follows from a result of Lih et al. ([2])described in the following.

For every vertex v ∈ V let L(v) be a set (list) of available colours for the vertex v.

Definition 9.2. A simple graphsG = (V ,E) is called (k,d)∗-list colourable if for everylist assignment with |L(v)| = k for all v ∈ V we can choose a colour ϕ(v) ∈ L(v)such that for every colour i the maximum degree of G[Vi] is at most d whereVi = {v ∈ V | ϕ(v) = i}.

Lih et al. ([2]) proved that every planar graph without 4- and `-cycles for some` ∈ {5, 6, 7} is (3, 1)∗-list colourable. Dong and Xu ([1]) proved an analogous resultfor planar graphs without 4- and `-cycles for some ` ∈ {8, 9}.On the other hand there are planar graphs without 4- and 5-cycles which are not3-list colourable ((3, 0)∗-list colourable) ([3]).

References:[1] W. Dong, B. Xu: A note on list improper coloring of plane graphs, Discrete Applied Mathematics 157 (2009)

433-436

[2] K.-W. Lih, Z. Song, W. Wang, K. Zhang: A Note on List Improper Coloring Planar Graphs, Applied MathematicsLetters 14 (2001) 269-271

[2] M. Voigt: A non-3-choosable planar graph without cycles of length 4 and 5, Discrete Mathematics, 307 (2007),1013-1015

10

Colourings

10 [1, 1, t]-Colourings of Complete Graphs

(Arnfried Kemnitz, Massimiliano Marangio)Given non-negative integers r, s, and t, an [r, s, t]-colouring of a graph G =

(V(G),E(G)) is a mapping c from V(G) ∪ E(G) to the colour set {0, 1, . . . , k − 1}such that |c(v) − c(v ′)| > r for every two adjacent vertices v, v ′, |c(e) − c(e ′)| > s

for every two adjacent edges e, e ′, and |c(v) − c(e)| > t for all pairs of incidentvertices and edges, respectively. The [r, s, t]-chromatic number χr,s,t(G) of G isdefined to be the minimum k such that G admits an [r, s, t]-colouring.

This is an obvious generalization of all classical graph colourings since c is a ver-tex colouring if r = 1, s = t = 0, an edge colouring if s = 1, r = t = 0, anda total colouring if r = s = t = 1, respectively. Therefore, χ1,0,0(G) = χ(G),χ0,1,0(G) = χ ′(G), and χ1,1,1(G) = χ ′′(G) where χ(G) is the chromatic number,χ ′(G) the chromatic index, and χ ′′(G) the total chromatic number of the graph G.

For complete graphs Kn on n vertices it holds

χ1,1,1(Kn) = χ′′(Kn) =

{n if n odd,n+ 1 if n even

and we proved (see [1] and [2])

χ1,1,2(Kn) =

n if n = 1,n+ 2 if n > 3 odd, n = 2, n = 6, or n = 8,n+ 3 if n = 4 or n > 10 even,

χ1,1,t(Kn) = 2n+ t− 2 for n > 2 and t > n,

χ1,1,t(Kn) = n+ 2t− 2 for n > 3 and t > 3 with⌊n

2

⌋− 1 6 t 6 n− 1.

Problem 10.1. Is it true that χ1,1,t(Kn) = n + 2t + u if t < n where u is a small(positive or negative) constant (depending on n and t)?

Problem 10.2. Determine χ1,1,t(Kn) for 3 6 t 6⌊n2

⌋− 2.

References:[1] A. Kemnitz, J. Lehmann, M. Marangio, [1, 1, 2]-Colorings of Complete Graphs

submitted.

[2] A. Kemnitz, M. Marangio, [r, s, t]-Colorings of GraphsDiscrete Mathematics 307 (2007), 199–207.

[3] A. Kemnitz, M. Marangio, Z. Tuza, [1, 1, t]-Colorings of Complete Graphs, submitted.

11

Colourings

11 Rainbow connection and size of graphs

(Ingo Schiermeyer)An edge-coloured connected graph G is called rainbow-connected if each pair ofdistinct vertices of G is connected by a path whose edges have distinct colours.The rainbow connection number of G, denoted by rc(G), is the minimum numberof colours such that G is rainbow-connected. In [1] the following problem wasintroduced.

Problem 11.1. For all integers n and k with 1 6 k 6 n− 1 compute and minimizethe function f(n,k) with the following porperty: If |V(G)| = n and |E(G)| > f(n,k)then rc(G) 6 k.

In [1] the following lower bound for f(n,k) has been shown.

Proposition 11.1. For n and k with 1 6 k 6 n−1 it holds that f(n,k) >(n−k+1

2

)+

k− 1.

This lower bound is tight which can be seen by construction of a graph Gk asfollows: Take a Kn−k+1 − e and denote the two vertices of degree n − k − 1with u1 and u2. Now take a path Pk with vertices labeled w1,w2, . . . ,wk andidentify the vertices u2 and w1. The resulting graph Gk has order n and size|E(Gk)| =

(n−k+1

2

)+ k − 2. For its diameter we obtain d(u1,wk) = diam(Gk) =

k+ 1 = rc(Gk).

Problem 11.2. Determine all values of n and k such that

f(n,k) =(n− k+ 1

2

)+ k− 1.

It has been shown that f(n,k) =(n−k+1

2

)+ k− 1 for

• k = 1, 2,n− 2, and n− 1 in [1],

• for k = 3 and 4 in [3],

• for n− 6 6 k 6 n− 3 in [2].

References:[1] A. Kemnitz, I. Schiermeyer, Graphs with rainbow connection number two

Disscussiones Math. Graph Theory 31 (2011), 313-320.

[2] A. Kemnitz, J. Przybylo, I. Schiermeyer, M. Wozniak, Rainbow connection in sparse graphsPreprint 2012.

[3] X. Li, M. Liu, I. Schiermeyer, Rainbow connection number of dense graphsPreprint 2011, Nankai University.

12

Colourings

12 Lower bounds of special edge colourings

(Stephan Matos Camacho)Consider a set H of binary vectors of length l. We define an addition h1 ⊕ h2 forelements h1,h2 ∈ H in the following way:

(h1 ⊕ h2)[i] :=

0 if h1[i] = h2[i] = 0,1 if h1[i] = h2[i] = 1,2 if h1[i] , h2[i].

Let G = (V ,E) be an edge-coloured graph, such that every v ∈ V corresponds toa vector h ∈ H, and two edges e, f ∈ E with e = v1v2 and f = v3v4 are colouredwith the same colour iff v1 ⊕ v2 = v3 ⊕ v4.

Problem 12.1. Determine rd(G), the minimum number of colours occurring insuch a graph G.

It can be easily computed, that rd(K2) = 1, rd(K3) = 3, rd(K4) = 5, rd(K5) = 9,rd(K6) = 12, rd(K7) = 16.

Conjecture 12.1.

rd(Kn) =

3k − 2k , if n = 2k,3k − 2k − k , if n = 2k − 1,3k , if n = 2k + 1.

This problem is motivated by a question concerning the Minimum Rainbow Sub-graph problem ([1])

References:[1] S. Matos Camacho, Introduction to the Minimum Rainbow Subgraph problem

PhD thesis, TU Bergakademie Freiberg (2012)

13

Colourings

13 Polychromatic colourings of cube graphs

(Heiko Harborth)The polychromatic numbers e(n,k) and g(n,k) denote the maximum numbers ofcolours for the edges and for the vertices, respectively, of the n-dimensional cubegraph such that each k-dimensional subcube graph contains all colours. See [1] forthe limits of e(n,k) and g(n,k) if k is fixed. - What about exact values of e(n,k) andg(n,k)? See [2] for some first exact values of g(n,k).

References:[1] Noga Alon, Anja Krech, and Tibor Szabo, Turan’s theorem in the hypercube

SIAM J. Discr. Math. 21 (2007), 66-72.

[2] Jens-P.Bode and Heiko Harborth, Polychromatic vertex colorings of cube graphsJ. Combin. Math. Combin. Comput. 80 (2012), 25-29.

14 Saturated rainbow edge colouring of

cube graphs

(Heiko Harborth, Arnfried Kemnitz)Let f(n,k) denote the minimum number of colours for the edges of the cube graphQn such that for k < n no rainbow Qk occurs (the edges of a rainbow Qk havepairwise different colours), however, for every edge with a colour used at leasttwice it follows that a new colour for this edge induces a rainbow Qk.For k = 3 it is known f(3, 3) = 11, f(4, 3) = 22, and f(5, 3) = 20.

1. Determine f(6, 3).

2. Determine the smallest n > 3 such that f(n, 3) = 11.

14

Colourings

15 Edge-distinguishing index of a sum of cycles

(Rafał Kalinowski, Mariusz Wozniak)A neighbourhood N(e) of an edge e of a graph G = (V ,E) is a subgraph ofG induced by e and all edges adjacent to e. A colouring c : E → S is callededge-distinguishing if, for any two distinct edges e, e ′, there does not exist anisomorphismϕ ofN(e) ontoN(e ′) preserving colours of c, and such thatϕ(e) = e ′.An edge-distinguishing index χ ′e(G) of a graphG is the minimum number of coloursin a proper edge-distinguishing colouring c : E→ S.If G is a cycle Cn of length n, then it is easy to see that

χ ′e(Cn) > γn := min{ k |12k2(k− 1) > n }.

Theorem 15.1.

χ ′e(Cn) =

{γn + 1 if n = 1

2k2(k− 1) − 1 or n = 4,

γn otherwise.

Problem 15.1. Let G be a disjoint sum of cycles with the total sum of lengths equalto n. Evaluate χ ′e(G).

15

Colourings

16 Neighbours distinguishing index of

planar graphs

(Keith Edwards, Mirko Hornak, Mariusz Wozniak)Let G be a finite simple graph with no component K2. Let C be a finite set ofcolours and let ϕ : E(G)→ C be a proper edge colouring of G. The colour set of avertex v ∈ V(G) with respect to ϕ, in symbols Sϕ(v), is the set of colours of edgesincident with v. The colouring ϕ is neighbours distinguishing if Sϕ(x) , Sϕ(y) forany xy ∈ E(G). For example, any neighbour-distinguishing colouring of C5 usesnecessarily 5 colours.The neighbours distinguishing index of the graph Gis the smallest number ndi(G)of colours in a neighbour-distinguishing colouring of G. Neighbours distinguishingindex has been introduced in [7], where the authors have conjectured that ndi(G) 6∆(G)+2 for any connected graphG nonisomorphic to C5 on at least three vertices(Neighbour-Distinguishing Conjecture = NDC). NDC was confirmed in [1] forcubic graphs and for bipartite graphs, in [6] for graphs with maximum degree atmost 3, in [2] for planar graphs with girth at least 6 and in [5] for planar graphswith maximum degree at least 12. In [3] it was proved that ndi(G) 6 ∆(G) + 1for any planar bipartite graph G with ∆(G) > 12. Hatami in [4] showed thatndi(G) 6 ∆(G) + 300 provided that ∆(G) > 1020.

Problem 16.1. Find the minimum integer ∆ > 4 such that ndi(G) 6 ∆(G) + 1 forany plane bipartite graph G with ∆(G) > ∆.

Problem 16.2. Prove or disprove NDC for planar graphs G with ∆(G) = ∆ for (atleast some) ∆ ∈ {4, . . . , 11}.

References:[1] P. N. Balister, E. Gyori, J. Lehel and R. H. Schelp, Adjacent vertex distinguishing edge-colorings

SIAM J. Disc. Math. 21 (2007), 237-250.

[3] Y. Bu, K.W. Lih and W. Wang, Adjacent vertex distinguishing edge-colorings of planar graphs with girth at least sixsubmitted.

[3] K. Edwards, M. Hornak and M. Wozniak, On the neighbour-distinguishing index of a graphGraphs Combin. 22 (2006), 341-350.

[4] H. Hatami, ∆+ 300 is a bound on the adjacent vertex distinguishing edge chromatic numberJ. Comb. Theory Ser. B 95 (2005), 246-256.

[5] M. Hornak, On neighbour-distinguishing index of planar graphsmanuscript.

[6] W. Wang, Y. Wang, Adjacent vertex distinguishing edge-colorings of graphs with smaller maximum average degreeJ. Comb. Optim. 19 (2010), 471-485.

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

16

Colourings

17 Edge colourings and the number of palettes

(Mirko Hornak, Rafał Kalinowski, Mariusz Wozniak)Let G be a finite simple graph of order n. Let C be a finite set of colours and letϕ : E(G) → C be a proper edge colouring of G. The palette of a vertex v ∈ V(G)(with respect to ϕ) is the set of colours of edges incident with v. The numberof (distinct) palettes ranges between 1 and n and any colouring that uses |E(G)|

colours produces the maximum number n of palettes (except for graphs havingtwo isolated vertices or containing K2 as a component). So, we are interested inthe minimum number of palettes taken over all possible proper colourings of G.We call this number the palette index of G and denote it by s(G).

Known results: We have proved in [1]:• s(G) = 1⇔ G regular, class 1• n ≡ 3 (mod 4)⇒ s(Kn) = 3• n ≡ 1 (mod 4)⇒ s(Kn) = 4• G connected, cubic, class 2, having a perfect matching ⇒ s(G) = 3• G connected, cubic, class 2, having no perfect matching ⇒ s(G) = 4

Thus Rathen Problem 14.1/2011 has been completely solved, the sequence {s(Kn)}∞n=1is bounded.

Problem 17.1. Determine s(Kp×q) for the complete p-partite graph with all partsof cardinality q.

References:[1] M. Hornak, R. Kalinowski, M. Meszka, M. Wozniak, Minimum number of palettes in edge colorings

manuscript.

17

Colourings

18 The neighbour-distinguishing-index by sums

in total proper colourings

(Monika Pilsniak, Mariusz Wozniak)Let c : V ∪ E → {1, 2, ..., k} be a proper total coloring of a graph G. For a vertexv, we denote by f(v) the sum of colors of the edges incident to v and of the colorof v. The smallest k that guarantees that there is a coloring c, so that the functionf distinguishes adjacent vertices of G, is called the total-neighbour-distinguishing-index by sums of G, and it is denoted by tndi∑(G).If we consider either trees and regular bipartite graphs, then tndi∑(G) equalsto ∆ + 1, if there does not exist two adjacent vertices of maximum degree, ortndi∑(G) = ∆+ 2, otherwise.For complete graphs, we can show that the total-neighbour-distinguishing-indexdepends on the parity of their order. Namely,

tndi∑(Kn) ={n+ 1, ifn is even,n+ 2, ifn is odd.

We can also show that, if G is a bipartite graph, or a cubic graph, or a graph with∆ 6 3, then tndi∑(G) 6 ∆+ 3. Hence, we may conjecture:

Conjecture 18.1. ([2]) For every graphG = (V ,E), the total-neighbour-distinguishing-index by sums satisfies the inequality

tndi∑(G) 6 ∆+ 3.

In [3], the authors investigated also a proper total coloring ofG, but for every vertexv they assigned a set S(v) of colors of the edges incident to v and the color of v.Similarly as above, by tndi(G) we can denote the smallest number k of colors, sothat there exists a proper total coloring c for G and S(u) is different from S(v) forevery pair of adjacent vertices u, v.

Conjecture 18.2. ([3]) For every graphG = (V ,E), the total-neighbour-distinguishing-index by sets tndi(G) satisfies the inequality

tndi(G) 6 ∆+ 3.

Zhang, Chen, Li, Yao, Lu and Wang considered the cases of cliques, paths, cycles,fans, wheels, stars, complete graphs, bipartite complete graphs and trees. Theyshowed that ∆+ 3 colors are enough in these cases. Next in [1] Chen proved thisconjecture for bipartite graphs and for graphs with maximum degree at most three.It is easy to observe, that if two vertices are distinguished by sums then they arealso distinguished by sets, but not necessarily conversely.

18

Colourings

References:[1] X. Chen, On the adjacent vertex distinguishing total coloring numbers of graphs with ∆ = 3

Discrete Mathematics Volume 308, Issue 17 (2008), 4003-4007.

[2] M. Pilsniak, M. Wozniak, On the adjacent-vertex-distinguishing index by sums in total proper coloringsPreprint MD 051, available at http://www.ii.uj.edu.pl/preMD/index.php.

[3] Z. Zhang, X. Chen, J. Li, B. Yao, X. Lu, J. Wang, On adjacent-vertex-distinguishing total coloring of graphsScience in China Ser. A Mathematics Vol. 48 No. 3 (2005), 289-299.

19

Cycles

19 Every locally connected graph is

weakly pancyclic

(Zdenek Ryjacek)Let G be a finite simple undirected graph and let g(G) and c(G) be the girth andthe circumference of G (i.e. the length of a shortest cycle of G and the length of alongest cycle of G), respectively. We say that G is weakly pancyclic if G containscycles of all lengths ` for g(G) 6 ` 6 c(G). The graph G is locally connected if theneighborhood of every vertex of G induces a connected graph.

Conjecture 19.1. ([8]) Every connected locally connected graph is weakly pancyclic.

Comments: The concept of locally connected graphs was introduced by Char-trand and Pippert ([3]). More information about weakly pancyclic graphs appearsin [2], for example.

The conjecture is based on a result by Clark ([4]), who proved that every connected,locally connected graph is vertex pancyclic (having cycles of all lengths from 3 to|V(G)| through every vertex). Without the claw-free assumption, it is easy to con-struct locally connected graphs that are nonhamiltonian. Nevertheless, all knownexamples are weakly pancyclic; and indeed [4] proved the conjecture for claw-freegraphs.

In a chordal graph, every block is locally connected, and for every cycle of length atleast 4 there is a cycle with length one less that is obtained by skipping one vertex.Thus the conjecture holds for chordal graphs.

It is easy to show that the square of any graph is locally connected. (The squareadds edges making vertices at distance 2 in the original graph adjacent.) Fleischner([5], Theorem 6) proved that the square of every graph is weakly pancyclic, thusverifying the conjecture for squares of graphs.

The lexicographical product of graphs is another way to obtain a locally connectedgraph. Kaiser and Kriesell ([6]) recently proved that the lexicographical productG[H] is weakly pancyclic provided G is a connected graph and H is an arbitrarygraph with at least one edge.

Kriesell ([7]) verified the conjecture for graphs with maximum degree at most 4.

20

Cycles

Finally, planar triangulations are locally connected. Balister ([1]) proved the con-jecture for this class as follows. Let C be a cycle in a planar triangulation G. Byinduction on the number of faces inside, we prove that the interior (with boundary)contains cycles of all shorter lengths. If some face inside has two edges on C,then using the third edge yields a cycle C ′ with length one less and fewer facesinside. Otherwise, there is a face with one edge on C and the third vertex inside.Detouring from C to include this vertex forms a longer cycle C ′, but again it hasfewer regions inside and the induction hypothesis applies.

References:[1] P. Balister, personal communication.

[2] S. Brandt, R. Faudree , W. Goddard, Weakly pancyclic graphsJ. Graph Theory 27 (1998), 141-176.

[3] G. Chartrand, R. Pippert, Locally connected graphsCas. pro pest. mat. 99 (1974), 158-163.

[4] L. Clark, Hamiltonian properties of connected locally connected graphsCongr. Numer. 32(1981), 199-204.

[5] H. Fleischner, In the square of graphs, hamiltonicity and pancyclicity, hamiltonian connectedness and pancon-nectedness are equivalent conceptsMonatshefte fur Mathematik 82 (1976), 125-149.

[6] T. Kaiser, M. Kriesell, On the pancyclicity of lexicographic procductsGraphs Comb. 22 (2006), 51-58.

[7] M. Kriesell, personal communication.

[8] Z. Ryjacek, Weak pancyclicity of locally connected graphsProblem 416, Discrete Mathematics 272 (2003), 305-306.

21

Cycles

20 Dominating cycles and hamiltonian prisms

(Zdenek Ryjacek)The prism over a graph G, denoted G�K2, is the Cartesian product of G and K2. Itconsists of two disjoint copies of G and a perfect matching connecting a vertex inone copy of G to its “clone" in the other copy.

A graph G is hamiltonian if it has a hamiltonian cycle and traceable if it has ahamiltonian path. Define a k-walk in a graph to be a spanning closed walk in whichevery vertex is visited at most k times.

The following implications are easy to verify:

G is hamiltonian⇒ G is traceable⇒ G�K2 is hamiltonian⇒ G has a 2-walk.

Thus the question whether G�K2 is hamiltonian is “sandwiched” between hamil-tonicity and having a 2-walk. Specifically, the property of having a hamiltonianprism can be considered as a “relaxation" of hamiltonicity. More information aboutprism-hamiltonicity of a graph can be found e.g. in [1] and [2].

A dominating cycle in a graph G is a cycle C such that every edge of G has at leastone vertex on C, i.e. such that the graph G−C is edgeless. Clearly, a hamiltoniancycle is dominating, and hence the property of having a dominating cycle can beconsidered as another relaxation of hamiltonicity.

There is a natural question whether there is any relation between these two prop-erties.

Example 20.1. Let H be any 2-connected cubic nonhamiltonian graph, and let Gbe obtained from H by replacing every vertex of H with a triangle (such a G issometimes called the inflation of H). Then G is a 2-connected line graph andthese are known [2] to be prism-hamiltonian. On the other hand, since H isnonhamiltonian, any cycle in G has to miss at least one “new" triangle and hence Ghas no dominating cycle. Thus, there are “many" graphs showing that hamiltonianprism does not imply having a dominating cycle.

Example 20.2. The graph in the figure below shows that also the existence of adominating cycle does not imply having hamiltonian prism.

22

Cycles

However, all such known examples are of low toughness. This motivates thefollowing question.

Conjecture 20.1. Let G be a 1-tough graph having a dominating cycle. Then G hashamiltonian prism.

Comments: Recall that G is 1-tough if, for any S ⊂ V(G), the graph G− S has atmost |S| components.

Suppose that G has a dominating cycle C of even length. Set M = V(G) \ V(C)

and N = {x ∈ V(C)| x has a neighbor in M}. Then the graph induced by M ∪ Nhas a matching containing all vertices from M (this follows by the toughness as-sumption and by the Hall’s theorem). Using this matching, it is easy to constructa hamiltonian cycle in G�K2.

The difficult case is when all dominating cycles in G are of odd length.

References:[1] R. Cada, T. Kaiser, M. Rosenfeld, Z. Ryjacek, Hamiltonian decompositions of prisms over cubic graphs

Discrete Mathematics 286 (2004), 45-56.

[2] T. Kaiser, D. Kral, M. Rosenfeld, Z. Ryjacek H.J. Voss, Hamiltonian cycles in prisms over graphsJ. Graph Theory 56 (2007), 249-269.

23

Cycles

21 Nonpancylic claw-free graphs with

complete 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 pancyclicclosure [1]. The graph in the figure below is an example of such a nonpancyclicgraph with complete (and hence pancyclic) closure.

Problem 21.1. Determine the maximum number of cycle lengths that can be miss-ing in a claw-free graph on n vertices with complete closure.

It is easy to see that a claw-free graph with complete closure on at least 4 verticescan miss neither a C3 nor a C4. The main result of [2] shows that such a graphG cannot be missing a cycle of length n − 1; however, the proof of this result isdifficult and cannot be iterated.

The following was conjectured in [2].

Conjecture 21.1. Let c1, c2 be fixed constants. Then for large n, any claw-free graphG of order nwhose closure is complete contains cyclesCi for all i, where 3 6 i 6 c1and n− c2 6 i 6 n.

Recently, counterexamples to the first part of the Conjecture 21.1 have been found(see [3]), all these counterexamples have connectivity κ 6 5. We believe that thesecond part of Conjecture 21.1 is true, and that such a construction as shown in[3] is possible only for connectivity κ 6 5. Thus, we conjecture the following.

Conjecture 21.2. Let c be a fixed constand. then for large n, any claw-free graph Gof order n, whose closure is complete, contains cycles Ci for all i,n− c 6 i 6 n.

Conjecture 21.3. Every 6-connected claw-free graph with complete closure is pan-cyclic.

24

Cycles

References:[1] S. Brandt, O. Favaron, Z. Ryjacek, Closure and stable hamiltonian properties in claw-free graphs

J. Graph Theory 34 (2000), 30-41.

[2] Z. Ryjacek, A. Saito, R.H. Schelp, Claw-free graphs with complete closureDiscrete Math. 236 (2001), 325-338.

[3] Z. Ryjacek, Z. Skupien, P. Vrana, On cycle lengths in claw-free graphs with complete closuremanuscript 2006 (submitted).

22 Hamiltonian neighborhood graphs

(Martin Sonntag, Hanns-Martin Teichert)The neighborhood graph N(G) for a simple graph G = (V ,E) is defined to be thegraph on the same vertex set V with two vertices adjacent if and only if there is inG a path of length two between them. Neighborhood graphs, also referred to astwo-step graphs, have been the object of several studies in the last 25 years.

If G is hamiltonian then N(G) is hamiltonian for |V | odd. This is not true for |V |even, for instance N(C2n) is disconnected. We can show that N(G) is alwayshamiltonian if G is 1-hamiltonian connected and has a triangle, but we think thereare weaker conditions providing hamiltonicity of N(G).

Problem 22.1. Find sufficient conditions for G, such that N(G) is hamiltonian orhamiltonian connected.

25

Decompositions & Partitions

23 Arbitrarily vertex-decomposable trees

(Mirko Hornak, Antoni Marczyk and Mariusz Wozniak)A tree T is said to be arbitrarily vertex decomposable if for any sequence (t1, . . . , tk)of positive integers adding up to |V(T)| there is a sequence (T1, . . . , Tk) of vertex-disjoint subtrees of T such that |V(Ti)| = ti for i = 1, . . . ,k.The notion of an arbitrarily vertex decomposable (avd for short) tree has been in-troduced independently by Barth et al. in [1] and Hornak and Wozniak in [5].

It turned out that some classes are essential when analysing the property of a tree“to be avd”. A star-like tree (a spider) is a tree homeomorphic to a star K1,q. Such atree is uniquely (up to isomorphism) determined by the non-decreasing sequence(a1, . . . ,aq) of orders of its arms; it will be denoted by S(a1, . . . ,aq) and also calleda q-spider.A caterpillar is a tree T having as a subgraph a path P such that T −P is an edgelessgraph.

The most general result concerns the best upper bound avdmax on the maximumdegree of an avd tree. In [5] it has been conjectured that avdmax 6 6 and conjecturedthat avdmax = 4. Later it was shown that avdmax 6 5 ([7]) and avdmax 6 4 ([2]).More precisely, the result of [2] reads as follows:

Theorem 23.1. If a tree T is avd, then ∆(T) 6 4. Moreover, if a tree T is avd, theneach vertex of T of degree four is adjacent to a leaf.

Let us mention that there are avd trees with maximum degree 4, for exampleS(2, 2, 5, 7), hence, as conjectured, avdmax = 4.There are only few known families of avd trees. The following theorem has beenproved independently in [1] and [5] (see also [4] for another, much more compli-cated result of this type).

Theorem 23.2. A 3-spider S(2,a2,a3) is avd if and only if a2 and a3 are coprime.

For a1 > 2 let A2(a1) be the set of all a2’s such that a2 > a1 and there is a3 > a2such that S(a1,a2,a3) is avd. Similarly, for a2 > a1 > 2 let A3(a1,a2) be the setof all a3’s such that a3 > a2 and there is a4 > a3 such that S(a1,a2,a3,a4) is avd.From Theorem 23.1 it is clear that if A3(a1,a2) , ∅, then a1 = 2.We give below four “main" open questions concerning avd trees.

Question 23.1. Is A2(a1) , ∅ for all a1 > 2?

Question 23.2. Is A3(2,a2) , ∅ for all a2 > 2?

26

Decompositions & Partitions

Hornak and Wozniak ([6]) showed that A2(a1) , ∅ for all a1 ∈ {2, . . . , 28} andA3(2,a2) , ∅ for all a2 ∈ {2, . . . , 23}. According to [3] there are infinitely manya1’s such that A2(a1) , ∅.It is easy to see that an avd caterpillar has at most one vertex of degree four. InFigure 1 there is depicted an avd tree having two vertices of degree four.

24 72 36An edge labelled l stands for a subpath with l vertices of degree 2.

References:[1] D. Barth, O. Baudon and J. Puech, Decomposable trees: a polynomial algorithm for tripodes

Discrete Appl. Math. 119 (2002), 205-216.

[2] D. Barth and H. Fournier, A degree bound on decomposable treesDiscrete Math. 306 (2006), 469-477.

[3] D. Barth, H. Fournier and R. Ravaux, On the shape of decomposable treesDiscrete Math. 309 (2009), 3882-3887.

[4] S. Cichacz, A. Gorlich, A. Marczyk, J. Przybyło and M. Wozniak, Arbitrarily vertex decomposable caterpillars withfour or five leavesDiscuss. Math. Graph Theory 26 (2006), 291-305.

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

[6] M. Hornak and M. Wozniak, On arbitrarily vertex decomposable treesDiscrete Math. 308 (2008), 1269-1281.

[7] S. J. Rosenberg, P. Kovar, D. Froncek and K. Wimmer, Existence results for arbitrarily vertex decomposable treespreprint MD 006 (2004), http://www.ii.uj.edu.pl/preMD/index.htm.

27

Decompositions & Partitions

24 Vertex-disjoint independent sets

(Anja Kohl)While investigating the minimum order of k−chromatic Kr+1−free graphs the fol-lowing question concerning Ramsey numbers arises:

Question 24.1. Let G be a graph with clique number at most r, independencenumber 3, and order n = R(r+ 1, 3) + 1. Do there always exist two vertex-disjointindependent sets with 3 vertices?

Known:

• If the order of G is n = R(r+ 1, 3) + 2, then the answer is ”yes” ([1]).

• If G is a graph on at least 10 vertices then either G contains a clique or anindependent set on four vertices, or G contains two disjoint triangles ([2]).This affirms the question for r = 3.

References:[1] A. Kohl: "On the minimum order of k−chromatic Kr+1−free graphs"

Preprint 2010-01 (2010), TU Bergakademie Freiberg.

[2] N. Trotignon, "Structure des classes de graphes definies par l’exclusion de sous-graphes induits"Habilitation thesis (2009), Universite Paris.

28

Dominating Sets

25 Domination hypergraphs of tournaments

(Martin Sonntag, Hanns-Martin 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 Vand its edges are the dominating sets of cardinality two (see for instance [1]). Asa natural generalization the domination hypergraph DH(D) also has vertex set Vand 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 25.1. The domination hypergraphDH(Tn) of a tournament Tn is alwaysconnected.

The conjecture is true for n 6 9. We tested hundreds of bigger examples (up ton = 23) by Mathematica routines and found no counterexample. It is easy to provethat every nontrivial component of DH(Tn) contains at least three edges. A firststep to verify the conjecture could be the investigation of regular tournaments.

References:[1] Fisher, Lundgren, Merz, Reid, The domination and competition graphs of a tournament

J. Graph Theory 29 (1998), 103-110.

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

29

Properties of Graphs

26 Weight of graphs having a given property

(Stanislav Jendrol’)Let G be a graph of positive size and let e = xy be an edge of G. The weight of e isw(e) := degG(x) + degG(y) and the weight of G is w(G) := min(w(e) : e ∈ E(G)).Let n,m ∈ Z, n > 2, 1 6 m 6

(n2

), let P be a graph property and let P(n,m)

be the set of all graphs in P of order n and size m. If P(n,m) , ∅, we definew(n,m,P) := max(w(G) : G ∈ P(n,m)). In the case P = I (the most generalproperty “to be a graph”) the problem of determining w(n,m, I) was formulatedby Erdos during the conference in Prachatice held in 1990 and solved by Jendrol’and Schiermeyer in [1].

We have obtained the results mentioned below.

Let B denote bipartite graphs. For a pair (n,m) such that B(n,m) , ∅ leta,b, s,p,w be integers defined by a := dn−

√n2−4m2 e, b := dm

ae, s := ab − m,

p := min(s, 2) and w := a+ b− p.

Theorem 26.1. If n > 2 and 1 6 m 6 bn2cdn2e, then the following hold:

1. 1 6 w 6 w(n,m,B) 6 w+ 1 6 n.

2. If w(n,m,B) = w+ 1, then w(n,m,B) = n.

3. w(n,m,B) = n if and only if√n2 − 4m is an integer.

4. w(n,m,B) = n − 1 if and only if one of the numbers√n2 − 4m− 4 and√

(n− 1)2 − 4m is an integer.

5. If√n2 − 4m,

√n2 − 4m− 4 and

√(n− 1)2 − 4m are not integers, then

w(n,m,B) = w.

Let C denote connected graphs and D1+ graphs with minimum degree at least 1.For a pair (n,m) such that C(n,m) , ∅ let k,m ′, r, c,d, e be integers defined by(n2

)−(k+1

2

)< m 6

(n2

)−(k2

), m ′ :=

(n2

)−(k2

)−m, r :=

⌈m ′

n−k

⌉, c := 1 if either

m ′ 6 bn−k2 c or m ′ = (n − k − 1)2, c := 2 otherwise, d :=⌊2mn

⌋, e := 0 if either

m =(n2

)− 1 or d 6 n − 3 and 2m ≡ q (mod n), 0 6 q 6 d − 1, and e := 1

otherwise.

Theorem 26.2. If 48 6 n− 1 6 m 6(n2

)−(dn/2e

2

), k > n

2 and C ⊆ P ⊆ D1+, thenw(n,m,P) = 2n− k− r− c.

Theorem 26.3. If n > 15,(n2

)−(dn/2e

2

)+ 1 6 m 6

(n2

)and C ⊆ P ⊆ D1+, then

w(n,m,P) = 2d+ e.

30

Properties of Graphs

Problem 26.1. Complete the results of Theorem 27.2 for n 6 48 and those ofTheorem 26.3 for n 6 14.

Problem 26.2. Find w(n,m,P) for other graph properties P.

References:[1] S. Jendrol’, I. Schiermeyer, On a Max-Min problem concerning weights of edges

Combinatorica 21 (2001), 351-359.

27 Turan type numbers for edges in

hypercube graphs

(Heiko Harborth, Hauke Nienborg)Let f(n,k, s) denote the maximum number of edges of Qn such that at least sedges of every subgraph Qk are missing. Thus f(n,k, 1) is the Turan type problemof P. Erdos for subgraphs Qk of the host graph Qn. – Especially, determinef(n,k,k2(k−1) − 1) = E(n,k) being the maximum number of edges of Qn suchthat every subgraph Qk contains at most one edge.

References:[1] N. Alon, A. Krech, T. Szabo, Turan’s theorem in the hypercube

SIAM J. Discr. Math. 21 (2007),66-72.

[2] H. Harborth, H. Nienborg, Turan numbers relative to platonic solid graphsCongr. Numer. 157 (2002), 5-20.

28 Minimum number of crossings of pseudo

diagonals in convex polygons

(Heiko Harborth)Consider convex n-gons with all diagonals as pseudoline segments. Two diagonalswith a common vertex do not intersect and two disjoint diagonals intersect at mostonce. Determine the minimum number c(n) of crossings if multiple crossings areallowed. So, c(7) = 29, c(8) 6 49, c(9) 6 87, c(10) 6 126.

31

Paths

29 Multiple crossings in drawings of

complete graphs

(Heiko Harborth)LetM(2m) denote the maximum number of m-fold crossings of edges in drawingsof the complete graphK2m (at most one crossing for disjoint edges and no crossingsfor edges with a common vertex). Prove that M(2m) 6 2 for m > 5.

References:[1] H. Harborth, Drawings of graphs and multiple crossings

Graph Theory and its Applications to Algorithms and Computer Science. Edited by Y. Alavi et al. (Kalamazoo1984). John Wiley & Sons 1985, 413-421.

30 Minimum k-path vertex cover

(Jan Katrenic, Ingo Schiermeyer, Gabriel Semanisin)Let G be a graph and k > 2 be an arbitrary, fixed integer. Then S ⊂ V(G) is calleda k-path vertex cover of G if every path on k vertices in G contains a vertex from S.The cardinality of a minimum k-path vertex cover of G is denoted by ψk(G). Notethat ψ2(G) is the cardinality of a minimum vertex cover of G.The concept of k-path vertex cover is related to the algorithm of M. Novotny for asecure communication in wireless sensor networks (see [4]). The invariant ψk(G)was introduced in [1] and some exact values and estimations of it are presented in[1], [2]. For example, for d-regular graphs we have the following result:

Theorem 30.1. Let k > 2 and d > 1 be positive integers. Then for any d-regulargraph G the following holds:

ψk(G) >d− k+ 22d− k+ 2

|V(G)|.

An associated k-Path Vertex Cover Problem (abbreviated by k-PVCP) can be formu-lated in the following way:

Problem 30.1. Given a graph G and a positive integer k. Find a minimum k-pathvertex cover of G.

Some algorithmic aspects of the k-PVCP were studied in [1]. It was proved therethat the k-PVCP is NP-hard for any fixed integer k > 2. On the other hand, the

32

Paths

value of ψk(G) for trees and ψ3(G) for outerplanar graphs can be computed inlinear time. One can easily see, that for an arbitrary positive integer k, there is atrivial k-approximation of k-PVCP. The following non-deterministic improvementis obtained in [3]:

Theorem 30.2. For the 3-PVCP there is a deterministic algorithm DPVC of approxi-mation ratio max(2, 5

2 ·d∆−1

2 ed∆+1

2 e), which runs in time O(2∆nO(1)) on a graph G with n

vertices and maximum degree ∆.

A deterministic approach was presented in [6] and [7]:

Theorem 30.3. There is a deterministic 2-approximation algorithm for 3-PVCP.

Some algorithms for a modification of the original problem were studied in [5]. Thepresented results lead us to the following problem:

Problem 30.2. Let k > 4. Is there a k − 1 factor approximation algorithm for thek-PVCP?

References:[1] B. Bresar, F. Kardos, J. Katrenic, G. Semanisin, Minimum k-path vertex cover

Discrete Applied Mathematics 159 (2011) 1189–1195.

[2] B. Bresar, M. Jakovac, J. Katrenic, G. Semanisin, A. Taranenko, On the Vertex k-Path Coversubmitted.

[3] F. Kardos, J. Katrenic, I. Schiermeyer, On computing the minimum 3-path vertex cover and dissociation numberof graphsTheoretical Computer Science 412 (2011) 7009–7017.

[4] M. Novotny, Design and Analysis of a Generalized Canvas ProtocolInformation Security Theory and Practices. Security and Privacy of Pervasive Systems and Smart Devices, LNCS6033 (2010) 106–121.

[5] X. Liu, H. Lu, W. Wang, W. Wu, PTAS for the minimum k-path connected vertex cover problem in unit disk graphsJournal of Global Optimization (2011) 1–10.

[6] J. Tu, W. Zhou, A factor 2 approximation algorithm for the vertex cover P3 problemInformation Processing Letters 111 (2011) 683–686.

[7] J. Tu, W. Zhou, A primal-dual approximation algorithm for the vertex cover P3 problem

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