Fifth International Conference on Combinatorics, Graph ... · FIFTH INTERNATIONAL CONFERENCE ON COMBINATORICS, GRAPH THEORY AND APPLICATIONS, MARCH 16 – 20, 2015, ELGERSBURG Dear

Fifth International Conference on

Combinatorics, Graph Theory andApplications

March 16 – 20, 2015

Elgersburg

Technische Universitat Ilmenau

FIFTH INTERNATIONAL CONFERENCE ON COMBINATORICS, GRAPH THEORY AND

APPLICATIONS, MARCH 16 – 20, 2015, ELGERSBURG

Dear guests,

Welcome to the Fifth International Conference on Combinatorics, Graph Theory and Ap-plications in Elgersburg (March 16 – 20, 2015) which is to celebrate the 60th birthdays ofJochen Harant and Michael Stiebitz.

This conference brings together mathematicans of distinct parts of the world interested ingraph theory, combinatorics and their interactions. It succeeds the Elgersburg conferencesof 1996, 2000, 2009, and 2011.

The scientific programme consists of seven invited lectures and of many contributed talks.We are proud to welcome as our invited speakers:

Jørgen Bang-Jensen (University of Southern Denmark, Odense, Denmark)

Stanislav Jendrol’ (P.J. Safarik University, Kosice, Slovakia)

Alexandr V. Kostochka (University of Illinois at Urbana-Champaign, USA)

Zdenek Ryjacek (University of West Bohemia, Pilsen, Czech Republic)

Bjarne Toft (University of Southern Denmark, Odense, Denmark)

Zsolt Tuza (Hungarian Academy of Sciences, Budapest and University ofPannonia, Veszprem, Hungaria)

Mariusz Wozniak (AGH University of Science and Technology, Krakow, Poland)

We hope that your visit to Elgersburg is memorable and profitable. We wish you a pleasantstay in Thuringia. Enjoy the talks, problems, and discussions.

Matthias KriesellAnja PruchnewskiJens Schreyer



Invited talks

Jørgen Bang-Jensen (Odense): Antistrong digraphs

Stanislav Jendrol’ (Kosice): Edge colourings of plane graphs

Alexandr V. Kostochka (Urbana): Color-critical graphs cannot be too sparse

Zdenek Ryjacek (Pilsen): Closure techniques for highly hamiltonian graphs

Bjarne Toft (Odense): Interval edge-colourings - history, results and problems

Zsolt Tuza (Budapest, Veszprem): Coloring and domination

Mariusz Wozniak (Krakow): On neighbour distinguishing colourings from lists

Contributed talks

Jens-P. Bode (Braunschweig): Sets of Van der Waerden Tuples

Mieczysław Borowiecki(Zielona Gora):

On H-stable Graphs

Andreas Brandstadt (Rostock): News About Efficient Domination for F -Free Graphs

Christoph Brause (Freiberg): A Lower Bound on the Independence Number of a Graphin terms of Degrees and Local Clique Sizes

Czilla Bujtas (Veszprem): On the 2-domination number

Ewa Drgas-Burchardt(Zielona Gora):

Harmonious colourings of graphs

Igor Fabrici (Kosice): On total colorings of plane graphs

Christoph Helmberg(Chemnitz):

The Laplacian Energy of Threshold Graphs

Winfried Hochstattler (Hagen): Towards a flow theory for the dichromatic number

Premysl Holub (Plzen): Packing chromatic number in outerplanar graphswith maximum degree 3

Mirko Hornak (Kosice): On the palette index of a graph



Contributed talks

Rafał Kalinowski (Krakow): Distinguishing Cartesian products of graphsby edge colourings

Arnfried Kemnitz (Braunschweig): Sum list edge colorings of graphs

Ekkard Kohler (Cottbus): Linear time LexDFS on Cocomparability Graphs

Christian Lowenstein (Ulm): The Fano Plane and the Strong Independence Ratioin Hypergraphs of Maximum Degree Three

Tomas Madaras (Kosice): The structural properties of non-polyhedral plane graphs

Massimiliano Marangio(Braunschweig):

Sum List Colorings of Small Graphs

Dirk Meierling (Ulm): Cycle Lengths of Hamiltonian P`-free Graphs

Mariusz Meszka (Krakow): Palette index of complete multipartite graphs

Monika Pilsniak (Krakow): The distinguishing index of the Cartesian productof countable graphs

Jakub Przybyło (Krakow): Locally irregular graph colourings

Robert Scheidweiler (Aachen): Ehrhart polynomials and the Erdos MultiplicationTable Problem

Ingo Schiermeyer (Freiberg): Chromatic number of P5-free graphs

Jens M. Schmidt (Ilmenau): Computing Tutte Cycles

Erika Skrabul’akova (Kosice): On the number of edges in 1-planar bipartite graphs

Martin Sonntag (Freiberg): Competition structures of products of digraphs

Roman Sotak (Kosice): Describing 3-paths in graphs with bounded maximumaverage degree

Eckard Steffen (Paderborn): Nowhere-zero flows

Eberhard Triesch (Aachen): On lower bounds for the k-independence number of graphs

Carol Zamfirescu (Dortmund): Hypohamiltonian and almost hypohamiltonian graphs

Tudor Zamfirescu (Dortmund): Discs and other miscreants held in cages



Further Participants

Georg Ehnert (Ilmenau)

Jochen Harant (Ilmenau)

Erhard Hexel (Ilmenau)

Peter John (Ilmenau)

Matthias Kriesell (Ilmenau)

Anja Pruchnewski (Ilmenau)

Sebastian Richter (Chemnitz)

Horst Sachs (Ilmenau)

Jens Schreyer (Ilmenau)

Michael Stiebitz (Ilmenau)

Margit Voigt (Dresden)



Antistrong digraphsJØRGEN BANG-JENSEN

Department of Mathematics and Computer ScienceUniversity of Southern Denmark, Odense DK-5230, Denmark

(joint work with Stephanne Bessy, Bill Jackson and Matthias Kriesell)

An antidirected trail in a digraph is a trail (a walk with no arc repeated) in which thearcs alternate between forward and backward arcs. An antidirected path is an antidirectedtrail where no vertex is repeated. We show that one can decide in linear time whetherthey are connected by an antidirected trail. A digraph D is antistrong if it contains anantidirected (x, y)-trail starting and ending with a forward arc for every choice of x, y ∈V (D). We show that antistrong connectivity can be decided in linear time. We discussrelations between antistrong connectivity and other properties of a digraph and show thatthe arc-minimal antistrong spanning subgraphs of a digraph are the bases of a matroidon its arc-set. We show that one can determine in polynomial time the minimum numberof new arcs whose addition to D makes the resulting digraph the arc-disjoint union of kantistrong digraphs. In particular, we determine the minimum number of new arcs whichneed to be added to a digraph to make it antistrong. We use results from matroid theoryto characterize graphs which have an antistrong orientation and give a polynomial timealgorithm for constructing such an orientation when it exists. This immediately givesanalogous results for graphs which have a connected bipartite 2-detachment.

Keywords: antidirected path, bipartite representation, matroid, detachment, anticonnecteddigraph



Sets of Van der Waerden TuplesJENS-P. BODE

(Technische Universitat Braunschweig, Germany)

joint work with H. Harborth

A d-dimensional box Ba1,...,ad is defined as the set of lattice points (x1, . . . , xd) of Zd with1 ≤ xi ≤ ai for i = 1, . . . , d. An arithmetic progression of length l is a set of l collinearequidistant lattice points. Then a van der Waerden set Wd,c,l will be defined as the set ofall ‘smallest’ tuples (a1, . . . , ad) having the property that in each coloring of the pointsof Ba1,...,ad with c colors there exists a monochromatic arithmetic progression of lengthl. This generalizes the classical van der Waerden numbers which correspond to the cased = 1. For d ≥ 2 some van der Waerden sets will be determined.



On H-stable GraphsMIECZYSŁAW BOROWIECKI

Faculty of Mathematics, Computer Science and EconometricsUniversity of Zielona Gora

[email protected]

We will consider finite simple graphs.Definition ((H, k)-stability) ([1]). Let H be a fixed graph. If the graph G has the propertythat removing any k edges of G, resulting graph still contains a subgraph isomorphic toH , then we say that G is (H, k)-stable.An (H, k)-stable graph G is minimal if every proper subgraph of G is not (H, k)-stable.Let us denote by Stab(H, k) the set of all minimal (H, k)-stable graphs.For a fixed H and k, let

s(H, k) = min{|E(G)| : G ∈ Stab(H, k)},S(H, k) = max{|E(G)| : G ∈ Stab(H, k)}.

In [1] authors found the formula s(P4, k) as function of k.

I the talk will be presented:

• some lower and upper bounds for S(H, k),

• all elements of Stab(H, 1) for selected H ,

• relations of (H, k)-stability with a few well-known concepts in graph, hypergraphor matroid theory,

• some open problems.

References

[1] I. Horvath, G.Y. Katona, Extremal P4-stable graphs, Discrete Applied Mathematics,159 (2011) 1786–1792.



News About Efficient Domination for F -Free GraphsANDREAS BRANDSTADT

Universitat Rostock(joint work with Elaine Eschen, West Virginia University, Morgantown, WV, U.S.A., and

Erik Friese, University of Rostock, Germany)

The NP-complete EXACT COVER problem for a hypergraph H = (V, E) asks for theexistence of a subset F ⊆ E of hyperedges covering every vertex of V exactly once.The NP-complete EFFICIENT DOMINATION (ED) problem for a graph G = (V,E) cor-responds to the Exact Cover problem for the closed neighborhood hypergraph of G. Itis known that the ED problem is NP-complete for claw-free graphs, for bipartite graphsas well as for chordal graphs. Thus, ED is NP-complete for F -free graphs whenever Fcontains an induced subgraph isomorphic to claw or a chordless cycle Ck with k verti-ces, k ≥ 3. If F is claw-free and cycle-free then F is the disjoint union of paths (calleda linear forest). From a standard reduction, it follows that ED is even NP-complete for2P3-free chordal graphs and thus also for P7-free chordal graphs. For P6-free graphs, thecomplexity of ED is unknown; for all other linear forests F , ED is either NP-complete orsolvable in polynomial time.

Our new results are the following:

1. For P5-free graphs, where ED is solvable in polynomial time, it was an open pro-blem whether ED can be solved in linear time. Using modular decomposition, wefound a linear time algorithm for these graphs.

2. It is well known that the ED problem on a graph G = (V,E) can be reduced to theMaximum Weight Independent Set (MWIS) problem on the square G2 = (V,E2)with xy ∈ E2 iff the G-distance of x and y is at most 2. Using this reduction, wehave shown that for P6-free chordal graphs G, G2 is chordal and thus, ED is sol-vable in polynomial time for such graphs; this even holds for the larger class of(house,hole,domino)-free graphs. Moreover, using the Strong Perfect Graph Theo-rem by Chudnovsky, Robertson, Seymour, and Thomas, we have shown that squa-res of P6- and house-free graphs (P6- and bull-free graphs, respectively) are perfectgraphs. Since by a result of Grotschel, Lovasz and Schrijver, MWIS is solvable inpolynomial time, the ED problem is solvable in polynomial time for P6- and house-free graphs and for P6- and bull-free graphs. The complexity of ED for P6-freegraphs remains an open problem.



A Lower Bound on the Independence Number of a Graphin terms of Degrees and Local Clique Sizes

CHRISTOPH BRAUSE

TU Bergakademie Freiberg, Germany

Caro and Wei independently showed that the independence number α(G) of a graph G isat least

∑u∈V (G)

1dG(u)+1

. In this talk we conjecture the stronger bound

α(G) ≥∑

u∈V (G)

2

dG(u) + ωG(u) + 1

where ωG(u) is the maximum order of a clique ofG that contains the vertex u. We discussthe relation of our conjecture to recent conjectures and results concerning the indepen-dence number and the chromatic number. Furthermore, we prove our conjecture for somespecific graph classes.

The talk bases on a joint work with Hubert Randerath, Dieter Rautenbach, and IngoSchiermeyer.



On the 2-domination numberCSILLA BUJTAS

Department of Computer Science and Systems TechnologyUniversity of Pannonia, Veszprem, Hungary

In a graph G, a set D ⊆ V (G) is called k-dominating set, if each vertex not in D has atleast k neighbors in D. The k-domination number γk(G) is the minimum cardinality ofsuch a set D. This graph invariant was introduced by Fink and Jacobson in 1985.We give an algorithm for the construction of 2-dominating sets, which also yields upperbounds on the 2-domination number in terms of the minimum degree and the number ofvertices. Our proof technique uses a weight-assignment to the vertices where the weightsare changed according to a greedy-type procedure.



Harmonious colourings of graphsEWA DRGAS-BURCHARDT AND KATARZYNA GIBEK

Faculty of Mathematics Computer Science and Econometrics,University of Zielona Gora

email: [email protected]

A mapping g : V (G) → {1, . . . , λ} is a λ-harmonious colouring of a graph G, if itsatisfies the conditions:

1. g(v1) 6= g(v2) for each v1v2 ∈ E(G); and

2. {g(v1), g(v2)} 6= {g(x1), g(x2)} for any two different edges v1v2, x1x2 ∈ E(G).

The harmonious number of G , denoted h(G), is the least positive integer λ, such thatthere exists a λ-harmonious colouring of G [1, 2].

Let h(G, λ) denote the number of all λ-harmonious colourings of G. In this work weanalyse the expression h(G, λ) as a function of a variable λ. We observe that this is a po-lynomial in λ of degree |V (G)|. Moreover, we present a reduction formula for calculatingh(G, λ). Using reducing steps we show the meaning of some coefficients of h(G, λ) andprove the Nordhaus-Gaddum type theorem. Also the notions of harmonious equivalenceand uniqueness are discussed.

Keywords: Vertex colouring, graph polynomial, Nordhaus-Gaddum theorem

References

[1] O. Frank, F. Harary, M. Plantholt, The line-distinguishing chromatic number of agraph, Ars Combin. 14 (1982), 241–252.

[2] J. E. Hopcroft, M. S. Krishnamoorthy, On the harmonious coloring of graphs, SIAMJournal on Algebraic and Discrete Methods 4 (1983), 306-311.



On total colorings of plane graphsIGOR FABRICI

Institute of Mathematics, Pavol Jozef Safarik University in Kosice

(joint work with Stanislav Jendrol’)

The facial k-total-coloring of a connected plane graph G of minimum degree δ(G) ≥ 2is a coloring of its vertices and edges using k colors such that no two adjacent vertices,no two facially adjacent edges (i.e. consecutive edges on the boundary walk of a face ofG), and no incident vertex and edge receive the same color. We show that any connectedplane graph of minimum degree at least two has a facial 6-total-coloring. We provideexamples requiring five colors for this coloring. We determine the exact values of thefacial total chromatic number for plane triangulations with all vertices of even degreeand for maximal outerplane graphs, and we find tight upper bounds on the facial totalchromatic number for plane triangulations and for outerplane graphs.



The Laplacian Energy of Threshold GraphsCHRISTOPH HELMBERG

TU Chemnitz, Germany

joint work with: Vilmar Trevisan, Universidade Federal do Rio Grande do Sul, Brazil

The Laplacian energy of a graph is defined as the sum of the absolute values of the dif-ferences of average degree and eigenvalues of the Laplacian matrix of the graph. Thisspectral graph parameter is upper bounded by the energy obtained when replacing theeigenvalues with the conjugate degree sequence of the graph, in which the i-th numbercounts the nodes having degree at least i. Because the sequences of eigenvalues and con-jugate degrees coincide for the class of threshold graphs, these are considered likely can-didates for maximizing the Laplacian energy over all graphs with given number of nodes.We do not answer this open problem, but within the class of threshold graphs we give anexplicit and constructive description of threshold graphs maximizing this spectral graphparameter for a given number of nodes, for given numbers of nodes and edges, and forgiven numbers of nodes, edges and trace of the conjugate degree sequence in the generalas well as in the connected case. In particular this positively answers the conjecture thatthe pineapple maximizes the Laplacian energy over all connected threshold graphs withgiven number of nodes.



Towards a flow theory for the dichromatic numberWINFRIED HOCHSTATTLER

FernUniversitat in Hagen

In an attempt to tackle the 4-Color-Conjecture William T. Tutte developed the theoryof Nowhere-Zero-Flows which can be considered as a concept dual to colorings. Hisfamous 5-flow- and 4-flow-conjecture are still unsettled, while the 4-Color-Conjecturehas become a theorem.

Victor Neumann-Lara defined the dichromatic number of a digraph as the smallest num-ber of colors needed to color the vertices without introducing a monochromatic dicycle.He conjectured that any orientation of a simple planar graph has dichromatic number atmost 2. Dual to this we introduce the concept of Neumann-Lara flows. We discuss thepossibility that every three edge connected digraph admits a Neumann-Lara 2-flow.



Packing chromatic number in outerplanar graphswith maximum degree 3

NICOLAS GASTINEAU, PREMYSL HOLUB1, OLIVER TOGNI

1Department of Mathematics, University of West Bohemia, Plzen, Czech Republic

A packing k-colouring of a graph G is a mapping from the vertex set V (G) to the set{1, 2, . . . , k} (called colour set) such that any two vertices coloured with colour i are atdistance at least i + 1. Then the packing chromatic number χρ(G) of G is the smallestinteger l such that there is a packing l-colouring of G.

This concept was introduced by Goddard at el. in [2] under the name “broadcast colou-ring”, but then the name was changed to “packing colouring” by Bresar et al. in [1]. Sloperin [3] showed that a complete binary tree of arbitrary height at least three is packing 7-colourable, while the infinite complete ternary tree is not packing colourable (i.e., wecannot colour vertices of the infinite complete ternary tree with a finite number of co-lours). In [2], it is shown that for paths and cycles, the packing chromatic number is atmost 4. Hence is it a natural question which classes of graphs with maximum degree 3can be packing colourable (i.e., for which classes of graphs with ∆ = 3, the packingchromatic number is finite). Since the problem is very difficult even for the class of planargraphs with ∆ = 3, we study the class of outerplanar graphs with ∆ = 3. In this talkwe present some subclasses of outerplanar graphs with ∆ = 3, for which the packingchromatic number is finite.

References

[1] B. Bresar, S. Klavzar, D. F. Rall, On the packing chromatic number of Cartesianproducts, hexagonal lattice and trees, Discrete Appl. Math. 155 (2007) 2303–2311.

[2] W. Goddard, S. M. Hedetniemi, S. T. Hedetniemi, J. M. Harris, D. F. Rall,Broadcast chromatic numbers of graphs, Ars Combinatoria 86 (2008) 33–49.

[3] C. Sloper, Broadcast-coloring of trees, Reports in Informatics 233 (2002), 1–11.



On the palette index of a graphMIRKO HORNAK

Institute of Mathematics, P.J. Safarik UniversityJesenna 5, 040 01 Kosice, Slovakia

(with Juraj Hudak)

LetG be a graph,C a set of colours and ϕ : E(G)→ C a proper edge colouring ofG. Theϕ-palette of a vertex v ∈ V (G) is the set {ϕ(vw) : vw ∈ E(G)}. The ϕ-diversity is thenumber of distinct ϕ-palettes of vertices of G and the palette index of G is the minimumof ϕ-diversities over all proper edge colourings of G. Some results on the palette index ofcomplete bipartite graphs will be presented.



Edge colourings of plane graphsSTANISLAV JENDROL’

Institute of Mathematics, Pavol Jozef Safarik University in KosiceJesenna 5, 040 01 Slovakia

e-mail: [email protected]

Let A = {a, b, c, . . . } be a finite alphabet, whose elements are called letters (digits, colours,symbols, ...). The word of length n over A is an expression w = a1a2 . . . an, where ai ∈ A for alli = 1, 2, . . . , n. Subword w of the wordw is an expression w = aiai+1 . . . aj with 1 ≤ i ≤ j ≤ n.The cyclic word of length n is an expression w = a1a2 . . . an, n ≥ 2 (consider the cyclic word asa sequence of consecutive labels on the edges of a cycle of length n). A subword of a cyclic wordis its arbitrary part.A word is proper if no two consecutive letters in it are the same. The word a1a2 . . . an, n ≥ 1, israinbow if ai 6= aj for i 6= j. The word of the form a1a2 . . . a2k with property that ai = ai+k forall i = 1, 2, . . . , k is called the repetition. A word is called nonrepetitive if none of its subwordsis a repetition. A palindrom is any word which can be read in the same way from the front andfrom the back. The word is palindromfree if no its subword is a palindrom. A word is an oddone if at least one letter in it appears there an odd number of times. A word is a strong odd one ifeach used letter in it is used an odd numbers of times. A word is a unique maximum one if the“largest” letter in it appear exactly once.Consider a 2-connected plane graph. All its faces are bounded by cycles, called the facial cycles.If we label all the edges of a 2-connected plane graph G with letters from an alphabet A, then anyface α = [e1, e2, . . . , ek] determined by the edges e1, e2, . . . , ek can be associated with a cyclicword a1a2 . . . ak, where k is size (degree) of the face α and ai is a label of the edge ei. The worda1a2 . . . ak is called the facial word of the face α of G.In our talk we will consider the following problem:

Problem: What is the minimum number of letters in an alphabet A that allow to label the edgesof a given 2-connected plane graph G in such a way that all the facial words of G over A have agiven property P ?

We will give a survey on results and open questions concerning this problem for several propertiesof words. See, e.g. [1], [2], [3].

References

[1] Czap J., Jendrol’ S., Kardos F., Sotak R.: Facial parity edge colouring of plane pseudogra-phs, Discrete Math. 312 (2012), 2735–2740.

[2] Fabrici I., Jendrol’ S., Vrbjarova M.: Unique-maximum edge-colourings with respect tofaces of plane graphs, Discrete Appl. Math. (to appear).

[3] Havet F., Jendrol’ S., Sotak R., Skrabul’akova E.: Facial non-repetitive edge-colouring ofplane graphs, J. Graph Theory 66 (2011), 38–48.



Distinguishing Cartesian products of graphsby edge colourings

RAFAŁ KALINOWSKI

AGH University of Science and Technology30 Mickiewicza Av., 30-059 Krakow, Poland

The distinguishing index of a graph is the minimum number of colours in an edge colou-ring (not necessarily proper) that is not preserved by any nontrivial automorphism. Thisconcept is analogous to the distinguishing number introduced by Albertson and Collinsfor vertex colourings.

We prove that the distinguishing number of the r-th Cartesian power of a connected graphequals two for every r > 1 except for the cycle of length four. We also prove similarresults for the Cartesian product of two distinct graphs provided their orders and sizes donot differ two much.

This is joint work with Ola Gorzkowska and Monika Pilsniak.



Sum list edge colorings of graphs

Arnfried Kemnitz∗, Massimiliano Marangio, Techn. Univ. Braunschweig, GermanyMargit Voigt, University of Applied Sciences, Dresden, Germany

Let G = (V,E) be a simple graph, and for every edge e ∈ E let L(e) be a set (list) ofavailable colors. The graph G is called L-edge colorable if there is a proper edge coloringc of G with c(e) ∈ L(e) for all e ∈ E. A function f : E → N is called an edge choicefunction of G and G is said to be f -edge list colorable if G is L-edge colorable for everylist assignment L with |L(e)| = f(e) for all e ∈ E. Set size(f) =

∑e∈E

f(e) and define the

sum choice index χ′sc(G) as the minimum of size(f) over all edge choice functions f ofG.There exists a greedy coloring of the edges ofGwhich leads to the upper bound χ′sc(G) ≤12

∑v∈V d(v)2. A graph is called sec-greedy if its sum choice index equals this upper

bound.We present some general results on the sum choice index of graphs including a lowerbound and we determine this index for several classes of graphs. Moreover, we presentclasses of sec-greedy graphs as well as all such graphs of order at most 5.

Mathematics Subject Classification: 05C15

Keywords: edge coloring, list coloring, sum choice index



Linear Time LexDFS on Cocomparability Graphs

EKKEHARD KOHLER1 AND LALLA MOUATADID2

1 Brandenburg University of Technology, 03044 Cottbus, Germany,[email protected]

2 University of Toronto, Toronto ON M5S 2J7, Canada,[email protected]

Lexicographic depth first search, LexDFS for short, was introduced by Corneil and Krue-ger, and has already proved to be a powerful tool on cocomparability graphs. These algo-rithms use lexicographic depth first search as a preprocessing step on the graph and thenextend, or slightly modify the algorithms that solve the same problem on interval graphs,dispensing with the complement computation of the comparability graph, but also pre-senting near linear time solutions to some NP-complete problems, like Hamiltonian Path.The non-linearity is a result of the preprocessing step. We present the first linear timealgorithm to compute a LexDFS ordering of a cocomparability graph, therefore helpingachieve elegant linear time algorithms for this graph class.



Color-critical graphs cannot be too sparseALEXANDR V. KOSTOCHKA

Department of Mathematics

University of Illinois at Urbana-Champaign

Michael Stiebitz is one of the world renowned experts in color-critical graphs and hy-pergraphs. The goal of this talk is to survey the recent progress in lower bounds on thenumber of edges in k-critical n-vertex graphs and hypergraphs. Some applications of re-cent results are also discussed.



The Fano Plane and the Strong Independence Ratioin Hypergraphs of Maximum Degree Three

CHRISTIAN LOWENSTEIN

University of Ulm, Germany

A set S of vertices in a hypergraph H is strongly independent if no two vertices in Sbelong to a common edge. The strong independence number of H , denoted α(H), is themaximum cardinality of a strongly independent set in H . The rank of H is the size of alargest edge in H . Our main result is that the strong independence ratio of a hypergraphH with rank k and maximum degree 3 satisfies α(H)

n(H)≥ 3

7kand this bound is achieved for

all k ≡ 0 (mod 3). In particular, this bound is achieved for the Fano plane.



The structural properties of non-polyhedral plane graphsTHOMAS MADARAS

Institute of Mathematics, Pavol Jozef Safarik University in KosiceJesenna 5, 040 01 Slovakia

The local structure of plane graphs has been, within the decades of graph theory rese-arch, investigated in details both in the connection with applications in colourings as wellas in its own right; the various previously known results of this topic were, during lasttwo decades, unified and generalized using the approach of light graphs theory (recentlysurveyed in 2013 in the summary article by Jendrol’ and Voss). Note, however, that themajority of light graph results concern triangulations or polyhedral graphs, as droppingthese conditions yields a very limited positive results on the existence of light graphs incorresponding families of plane graphs. Our contribution shows how the extra conditions(involving mainly the weight of edges or size of faces) may enforce the existence of non-trivial light graphs even in the families of plane graphs of minimum degree at least 2; theparticular attention is also paid on the mutual relation of these conditions and determiningthose ones which lead to extremal subfamilies of non-polyhedral plane graphs.



Sum List Colorings of Small GraphsMASSIMILIANO MARANGIO

(Computational Mathematics, Technische Universitat Braunschweig)

LetG = (V,E) be a simple graph and for every vertex v ∈ V let L(v) be a list of availablecolors. G is called L-colorable if there is a proper vertex coloring c with c(v) ∈ L(v) forall v ∈ V . A function f : V → N is called a choice function of G if G is L-colorable forevery list assignment L with |L(v)| = f(v) for all v ∈ V . Set size(f) =

∑v∈V

f(v) and

define the sum choice number χsc(G) as the minimum of size(f) over all choice functionsf of G.A general upper bound for χsc(G) of a graph G is χsc(G) ≤ |V | + |E|. A graph G iscalled sc-greedy if χsc(G) = |V |+ |E|.In this talk the sum choice number of all graphs with at most six vertices is presented.This leads to a characterization which graphs are sc-greedy and which are not for graphsfrom some specific graph classes, for example complete bipartite graphs and completemultipartite graphs.

This is joint work with Arnfried Kemnitz and Margit Voigt.



Cycle Lengths of Hamiltonian P`-free GraphsDIRK MEIERLING, DIETER RAUTENBACH

Institut fur Optimierung und Operations Research, Ulm University,Ulm, Germany

[email protected], [email protected]

At the 1999 conference “Paul Erdos and His Mathematics”, Jacobson and Lehel asked how smallthe cycle spectrum of a Hamiltonian graph satisfying some other conditions can be. Specifically,for an integer k with 3 ≤ k ≤ d1

2ne−1, they asked for the minimum cycle spectrum of a k-regularHamiltonian graph on n vertices.Note that Bondy [1] proved that all k-regular graphs of order n with k ≥ d1

2ne, except forKn/2,n/2, are pancyclic. Furthermore, 2-regular Hamiltonian graphs have exactly one cycle length.During the SIAM Meeting on Discrete Mathematics in 2002, Jacobson announced that he, Gould,and Pfender proved that every k-regular Hamiltonian graph G on n vertices has cycles of atleast ck

√n different lengths where ck > 0 for k ≥ 3. Milans et al. [2] strengthened the re-

sult by showing that every Hamiltonian graph with n vertices and m edges has cycles of at least√m− n− ln(m−n)− 1 different lengths. In [3] Marczyk and Wozniak determine the minimum

size of the cycle spectrum of Hamiltonian graphs in terms of their maximum degree. Recently,Muttel et al. [4] proved that cubic claw-free Hamiltonian graphs of order n > 12 have cycles of atleast 1

4n+ 3 different lengths where a claw is a complete bipartite graph K1,3.Naturally, the existence, lengths, and distribution of chords of a Hamiltonian cycle is importantfor the existence of further cycles. The conditions mentioned above, such as degree conditions,regularity conditions, conditions on the size, or claw-freeness, all imply the existence of manychords and partly also concern their lengths and distribution.We consider P`-freeness as another natural condition that forces many well distributed chords ina Hamiltonian cycle. For an integer ` at least 3, a graph is P`-free if it contains no path of order` as an induced subgraph. For an integer ` at least three, we prove that every Hamiltonian P`-freegraph G on n > ` vertices has cycles of at least 2

`n − 1 different lengths. For small values of `,we can improve the bound as follows. If 4 ≤ ` ≤ 7, then G has cycles of at least 1

2n− 1 differentlengths, and if ` is 4 or 5 and n is odd, then G has cycles of at least n− `+ 2 different lengths.

References

[1] J. A. Bondy, Pancyclic Graphs. J. Combin. Theory (B) 11 (1971), 80–84.

[2] K. G. Milans, F. Pfender, D. Rautenbach, F. Regen, D. B. West, Cycle spectra of Hamilto-nian graphs. J. Combin. Theory (B) 102 (2012), 869–874.

[3] A. Marczyk, M. Wozniak, Cycles in hamiltonian graphs of prescribed maximum degree.Discrete Math. 266 (2003), 321–326.

[4] J. Muttel, D. Rautenbach, F. Regen, T. Sasse, On the Cycle Spectrum of Cubic HamiltonianGraphs. Graphs Comb. 29 (2013), 1067–1076.



Palette index of complete multipartite graphsMARIUSZ MESZKA

AGH University of Science and Technology, Krakow, [email protected]

Let c : E(G) 7→ C be a proper edge coloring of a graph G, i.e., incident edges of G getdistinct colors. A palette of a vertex v ∈ V (G) with respect to c is the set Sc(v) of colorsof edges incident to v. Two vertices of G are distinguished by a coloring c if their palettesare distinct. Given a coloring c of G, the number of distinct palettes among all vertices ofG is denoted by pc(G). The minimum value of pc(G) taken over all possible proper edgecolorings of G is called the palette index and is denoted by s(G).Colorings of some classes of graphs with respect to the minimum number of paletteswill be discussed. In particular, the palette index of complete multipartite graphs will becompletely determined.



The distinguishing index of the Cartesian productof countable graphs

MONIKA PILSNIAK

AGH University of Science and Technologyal. Mickiewicza 30, 30-059 Krakow, Poland

e-mail: [email protected]

The distinguishing index D′(G) of a graph G is the least cardinal d such that G has anedge colouring with d colours that is preserved only by the trivial automorphism. Wederive some bounds for this parameter for infinite graphs. In particular, we investigatethe distinguishing index of the Cartesian product of countable graphs. We show that it isinfinite iff one of factors is finite and an other has an infinite distinguishing index.

We consider also proper edge colourings in this definition to define the distinguishingchromatic index of a graph and we obtain similar results for this parameter.Finally, we prove that D′(Q) = 2, where Q is the infinite dimensional hypercube.



Locally irregular graph colouringsJAKUB PRZYBYŁO

AGH University of Science and Technology, Krakow, Poland

(joint work with A. Raspaud and M. Wozniak)

It is well known that there are no irregular graphs, understood as simple graphs withpairwise distinct vertex degrees, except the trivial 1-vertex case. A graph invariant calledthe irregularity strength was thus introduced aiming at capturing a level of irregularityof a graph. Suppose that given a graph G = (V,E) we wish to construct a multigraphwith pairwise distinct vertex degrees of it by multiplying some of its edges. The least k sothat we are able to achieve such goal using at most k copies of every edge is denoted bys(G) and referred to as the irregularity strength of G. Alternately one may consider (notnecessarily proper) edge colourings c : E → {1, 2, . . . , k} with

∑e3u c(e) 6=

∑e3v c(e)

for every pair of distinct vertices u, v ∈ V . Then the least k which permits defining acolouring c with this feature equals s(G). Numerous papers have been devoted to studyon this graph invariant since the middle 80’s.On the other hand, there are many locally irregular graphs, i.e., those whose only ad-jacent vertices are required to have distinct degrees. Analogously as above one mightalso measure how far a given graph G = (V,E) is from being locally irregular. That is,how large k is needed in order to define an edge colouring c : E → {1, 2, . . . , k} with∑

e3u c(e) 6=∑

e3v c(e) for every pair of adjacent vertices u, v of G. It is believed thatalready k = 3 is sufficient for all graphs containing no isolated edges. This suspicion iscommonly referred to as the 1–2–3–Conjecture.During the talk we shall briefly outline most well known theorems and conjectures ofthe field, and sketch recent joint results (obtained with A. Raspaud and M. Wozniak) oncertain list extensions of the 1–2–3–Conjecture and its total version. These concern sparsegraphs with Mad(G) < 5

2investigated by means of the discharging method combined

with the algebraic approach of N. Allon known as Combinatorial Nullstellensatz.



Closure techniques for highly hamiltonian graphsZDENEK RYJACEK

(University of West Bohemia, Plzen)

A graph G is hamiltonian if G has a hamiltonian cycle, Hamilton-connected if G has ahamiltonian (x, y)-path for any x, y ∈ V (G), and, for k ≥ 0, k-Hamilton-connected ifG−X is Hamilton-connected for any X ⊂ V (G) with |X| = k.

It is well-known that the closure concept for hamiltonicity in claw-free graphs, whichturns a claw-free graph into a line graph of a triangle-free graph, preserves many graphproperties (of hamiltonian type), however, many properties stronger than hamiltonici-ty (such as Hamilton-connectedness or 1-Hamilton-connectedness) are not preserved.In the talk, we show ideas that allow to construct closure operations which preserveHamilton-connectedness and 1-Hamilton-connectedness, and still keep (some of) the “ni-ceproperties of the closure for hamiltonicity, namely, that the closure is a line graph ofa (multi-)graph with a special structure. As applications, we show some degree condi-tions for strong hamiltonian properties, we discuss pairs of forbidden subgraphs for theproperties, and we discuss consequences for the well-known conjectures by Matthews-Sumner/Thomassen.



Ehrhart Polynomials and the Erdos Multiplication TableProblem

ROBERT SCHEIDWEILER

RWTH Aachen University, Aachen, Germany

(joint work with Eberhard Triesch)

Let

Pm(n) :=

{n∏i=1

iαi

∣∣∣∣∣αi ∈ N0 andn∑i=1

αi = m

}be the set of products of m numbers from the set {1, . . . , n}. In 1955 Erdos posed theproblem of determining the order of magnitude of |P2(n)|. This so-called Erdos Multi-plication Table Problem was settled in 2008 by Ford in [2]. Koukoulopoulos determinedthe order of magnitude of |Pm(n)| in [3]. Recently, Darda and Hujdurovic [1] asked if|Pm(n)| is a polynomial in m of degree π(n) - the number of primes not larger than n.Motivated by this question we present and discuss a connection between Ehrhart Theoryand the Erdos Multiplication Table Problem.

References

[1] R. Darda and A. Hujdurovic, On Bounds for the Product Irregularity Strength ofGraphs, Graphs and Combinatorics (2014).

[2] K. Ford, The distribution of integers with a divisor in a given interval, Annals ofMath. (2) 168 (2008), 367–433.

[3] D. Koukoulopoulos, Localized factorizations of integers, Proc. Lond. Math. Soc.(3) 101 (2010), no. 2, 392–426.



Chromatic number of P5-free graphsINGO SCHIERMEYER(1,2)

(1) TU Bergakademie Freiberg, Germany(2) AGH Cracow, Poland

In this talk we study the chromatic number of P5-free graphs. Our work was motivated bythe following conjecture of Gyarfas.

Conjecture(Gyarfas’ conjecture [1])Let T be any tree (or forest). Then there is a function fT such that every T -free graph Gsatisfies χ(G) ≤ fT (ω(G)).

Gyarfas [1] proved this conjecture when T is a path Pk for all k ≥ 3 by showing χ(G) ≤(k − 1)ω(G)−1.

In 1998, Reed proposed the following Conjecture which gives, for any graph G, an up-per bound for the chromatic number χ(G) in terms of the clique number ω(G) and themaximum degree ∆(G).

Conjecture(Reed’s conjecture [2])Every graph G satisfies χ(G) ≤ dω(G)+∆(G)+1

2e.

Reed’s conjecture is still open in general. Our main result is that the conjecture holdsasymptotically for Pk-free graphs.

Theorem For every fixed ω ≥ 3 there exists n(ω) such that if G is a connected Pk-freegraph of order n ≥ n(ω) and clique number ω, then χ(G) ≤ dω(G)+∆(G)+1

2e.

References

[1] A. Gyarfas, Problems from the world surrounding perfect graphs. In: Proc. Int.Conf. on Comb. Analysis and Applications (Pokrzywna, 1985), Zastos. Mat. 19(1987) 413–441.

[2] B. Reed, ω,∆ and χ. In: Journal of Graph Theory 27 (1998) 177–212.



Computing Tutte CyclesJENS M. SCHMIDT

Institut fur Mathematik, Technische Universitat Ilmenau

Tutte cycles build a powerful tool for proving structural results on planar graphs. Oneparticular implication of them is the existence of 2-walks in 3-connected graphs: A 2-walkof a graph is a walk visiting every vertex at least once and at most twice. Gao, Richterand Yu proved that every 3-connected planar graph contains a closed 2-walk such that allvertices visited twice are contained in 3-separators. For both, Tutte cycles and 2-walks,the algorithmic challenge is to overcome big overlapping subgraphs in the decomposition,which are also inherent in Tuttes and Thomassens decompositions. We solve this problemby extending the decomposition of Gao, Richter and Yu in such a way that all pieces, inwhich the graph is decomposed into, are edge-disjoint. This implies the first polynomial-time algorithm that computes the closed 2-walk mentioned above.



On the number of edges in 1-planar bipartite graphsERIKA SKRABUL’AKOVA

Institute of Control and Informatization of Production ProcessesFaculty of Mining, Ecology, Process Control and Geotechnology

Technical University of Kosice, Kosice, Slovakia

(joint work with Julius Czap and Jakub Przybyło)

A graph G = (V,E) is called 1-planar if it admits a drawing in the plane such that eachedge is crossed at most once. We study bipartite 1-planar graphs with prescribed numbersof vertices in partite sets. These are known to have at most 3n− 6 edges, where n denotesthe order of a graph. We reveal that maximal-size bipartite 1-planar graphs which arealmost balanced have not significantly fewer edges than indicated by this upper bound,while the same is not true for unbalanced ones. We show that maximal possible sizes ofbipartite 1-planar graphs whose one partite set is much smaller than the other one tendstowards 2n rather than 3n.



Competition structures of products of digraphsMARTIN SONNTAG

Technische Universitat Bergakademie Freiberg, D–09596 Freiberg, Pruferstr. 1(joint work with HANNS-MARTIN TEICHERT, Universitat zu Lubeck)

If D = (V,A) is a digraph, its competition hypergraph (with loops) CHl(D) has vertexset V and non-empty subset e ⊆ V is a hyperedge of CHl(D) iff there is a vertex v ∈ V ,such that e = {w ∈ V |(w, v) ∈ A}. For five products D1 ◦D2 of digraphs D1 and D2, weinvestigate the relations between the competition hypergraphs of the factors D1, D2 andthe competition hypergraph of their product D1 ◦D2.Moreover, some remarks to the analogous problem for competition graphs will be given.



Describing 3-paths in graphs with bounded maximumaverage degree

ROMAN SOTAK

P.J. Safarik University, Kosice, Slovakia

In this talk, we study the existence of paths on 3 vertices with given degree sequence insparse graphs. A path of type (x, y, z) (x, y, z ∈ N∗) is a path uvw such that the degreeof u (resp. v, w) is at most x (resp. y, z). We find the lists of possible types of 3-pathsfor some classes of graphs. Specifically, every graph with minimum degree at least 2 andmaximum average degree strictly less than m has a path of one of the types

• (2,∞, 2), (5, 2, 5), (4, 3, 5), (2, 8, 3) if m = 154

;

• (2,∞, 2), (3, 2, 4), (3, 3, 3), (2, 5, 3) if m = 103

;

• (2, 2,∞), (2, 3, 4), (2, 5, 2) if m = 3 ;

• (2, 2, 13), (2, 3, 3), (2, 4, 2) if m = 145

;

• (2, 2, i), (2, 3, 2) if m = 3(i+1)i+2

for 4 ≤ i ≤ 7 ;

• (2, 2, 3) if m = 125

;

• (2, 2, 2) if m = 94.

We discuss also the optimality of these results. Finally we give similar lists for planargraphs with restricted girth.

This is joint work with Stanislav Jendrol’, Maria Macekova and Mickael Montassier.



Nowhere-zero flowsECKHARD STEFFEN (JOINT WORK WITH GIUSEPPE MAZZUOCCOLO)

Universitat Paderborn, Warburger Str. 100, 33098 Paderborn

An integer nowhere-zero k-flow on a graph G is an assignment of a direction and a valueof {1, . . . , (k − 1)} to each edge of G such that the Kirchhoff’s law is satisfied at everyvertex of G. A cubic graph G is bipartite if and only if it has a nowhere-zero 3-flow,and χ′(G) = 3 if and only if G has a nowhere-zero 4-flow. Seymour [2] proved that everybridgeless graph has a nowhere-zero 6-flow. So far this is the best approximation to Tutte’sfamous 5-flow conjecture, which is equivalent to its restriction to cubic graphs.

Conjecture 1 ([4]) Every bridgeless graph has a nowhere-zero 5-flow.

By a result of Kochol [1], it suffices to prove Conjecture 1 for cyclically 6-edge-connectedcubic graphs.A classical parameter to measure how far a cubic graph is from being 3-edge- colorable isits oddness. The oddness, denoted by ω(G), of a bridgeless cubic graph G is the minimumnumber of odd circuits in a 2-factor of G. Since G has an even number of vertices, ω(G)is necessarily even. Furthermore, ω(G) = 0 if and only if G has a nowhere-zero 4-flow. Itis easy to see that bridgeless cubic graphs with oddness 2 have a nowhere-zero 5-flow.Furthermore, a consequence of the main result in [3] is that cyclically 7-edge connectedcubic graphs with oddness at most 4 have a nowhere-zero 5-flow. The following is ourmain theorem, and it is a strengthening for both previous results.

Theorem 1.1 Let G be a cyclically 6-edge-connected cubic graph. If ω(G) ≤ 4, thenG has a nowhere-zero 5-flow.

References

[1] M. KOCHOL, Reduction of the 5-flow conjecture to cyclically 6-edge- connectedsnarks, J. Combin. Theory, Ser. B 90 (2004) 139–145

[2] P. D. SEYMOUR, Nowhere-zero 6-flows, J. Combin. Theory, Ser. B 30 (1981) 130–135

[3] E. STEFFEN, Tutte’s 5-flow conjecture for highly cyclically connected cubicgraphs, Discrete Math. 310 (2010) 385–389

[4] W. T. TUTTE, A contribution to the theory of chromatic polynomials, Canadian J.Math. 6 (1954) 80–91



Interval edge colourings - history, results and problemsBJARNE TOFT

Department of Mathematics and Computer ScienceThe University of Southern Denmark, DK- 5230 Odense M, Denmark

A proper edge colouring of a graph with colours 1, 2, 3, ... is called an interval colouring ifthe colours incident to each vertex form an interval of integers. A bipartite graph is (a,b)-biregular if every vertex in one part has degree a and every vertex in the other part hasdegree b. The notion of interval colouring was introduced by A.S. Asratian and R.R. Ka-melian (Yerevan University, Armenia) in 1987, motivated by the problem of finding com-pact school timetables. H.M. Hansen (University of Odense, Denmark) described in histhesis in 1992 another senario (obtained from Jesper Bang-Jensen): a school wishes toschedule parent-teacher consultations in timeslots so that each persons consultations oc-cur in consequtive slots. The conjecture that every (a,b)-biregular bipartite graph has aninterval colouring appeared in Hansen’s thesis.

In this talk the history of interval colourings will be presented, including new results,in particular by Carl Johan Casselgren (Linkping University, Sweden). For example thatany (3,6)-biregular bipartite graph has an interval 7-colouring. Variations include cyclicinterval-colourings and near-interval colourings.

The five basic (related) unsolved problems are:

1. Does a bipartite graph of maximum degree 11 always have an interval colouring?2. Does a bipartite graph of maximum degree at most 4 always have an interval colouring?3. Does a (3,4)-biregular bipartite graph always have an interval colouring?4. Does an (a,b)-biregular bipartite graph always have a cyclic interval max{a,b}-colouring?5. Does an (a,b)-biregular graph always have an interval colouring?



On lower bounds for the k-independence number of graphsEBERHARD TRIESCH, AACHEN

We are interested in lower bounds for the k-independence number of graphs in terms ofthe degrees. Following ideas of Murphy from 1991 for k = 1, we generalize his boundto arbitrary k and compare the bound to the well-known bounds of Hopkins-Staton, Jelen(k-residue) and Caro-Hansberg. There are cases where the generalized Murphy boundyields better estimates than all the other bounds mentioned before.

(Joint work with Michael Hoschek, Aachen).



Coloring and dominationZSOLT TUZA

MTA Renyi Institute for Mathematics, Budapest & University of Pannonia, Veszprem,Hungary

We discuss results and open problems on various kinds of colorings and dominating setsin graphs and hypergraphs.



On neighbour distinguishing colourings from listsMARIUSZ WOZNIAK

Department of Discrete Mathematics, AGH University, Krakow, Polande-mail: [email protected]

Let us consider a colouring of edges of a simple graph G = (V,E). Such a colouringdefines for each vertex x ∈ V a palette of colours, i.e., the multiset of colours of edgesincident with x. These palettes can be used to distinguish some vertices of a graph. Ananalogous problem can be considered also for vertex or total colourings. There are manyinvariants depending on the kind of colourings we consider, the way we distinguish thevertices, and so on.We shall see a few examples of such problems. We shall focus on the case when wedistinguish adjacent vertices and colourings are from lists. These examples will illustratevarious methods. Another method will be discussed in the Jakub Przybyło’s talk.



Hypohamiltonian and almost hypohamiltonian graphsCAROL T. ZAMFIRESCU

TU Dortmund

Consider a non-hamiltonian graphG and a non-negative integer k.G is k-hypohamiltonianif there exists a set W ⊂ V (G) of cardinality k such that G − w is non-hamiltonian forevery w ∈ W and G − v is hamiltonian for every v /∈ W . For k = 0 this coincides withthe definition of a hypohamiltonian graph, and for k = 1 we shall call the obtained fami-ly of graphs almost hypohamiltonian. There is a striking discrepancy between the casesk ∈ {0, 1} and the cases k ≥ 2. In this talk we discuss the former, with special emphasison the planar case.



Discs and other miscreants held in cagesTUDOR ZAMFIRESCU

TU Dortmund

This talk is about cut loci on convex surfaces. The cut loci are considered with respect tocompact sets. Even if the set is a single point, its cut locus can have infinite length. So,there is no hope for an upper bound. What about a lower bound? If the set M consistsof 1 or 2 points, we quickly see that the lower bound is 0. However, if card M = 3, andthe convex surface S has diameter 2, it seems that the cut locus C(M) of M cannot havelength less than 1. For infinite M , that lower bound is again 0. Can we establish a lowerbound in case M is finite and card M > 2 ? Yes, and we do this in the talk.

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Fifth International Conference on Combinatorics, Graph ... · FIFTH INTERNATIONAL CONFERENCE ON COMBINATORICS, GRAPH THEORY AND APPLICATIONS, MARCH 16 – 20, 2015, ELGERSBURG Dear