A note on a Vizing's generalized conjecture

In this note we give a generalized version of Vizing's conjecture concerning the distance domination number for the cartesian product of two graphs

Total-chromatic number and chromatic index of dually chordal graphs

Given a graph G and a vertex nu, a vertex u is an element of N(nu) is a maximum neighbor of nu if for all w is an element of N(nu) we have N(zu) subset of or equal to N(u), where N(nu) denotes the neighborhood of nu in G. A maximum neighborhood elimination order of G is a linear order nu(1), nu(2),..., nu(n) on its vertex set such that there is a maximum neighbor of v(i) in the subgraph G[v(1),..., v(i)]. A graph is dually chordal if it admits a maximum neighborhood elimination order. Alternatively, a graph is dually chordal if it is the clique graph of a chordal graph. The class of dually chordal graphs generalizes known subclasses of chordal graphs such as doubly chordal graphs, strongly chordal graphs, interval graphs, and indifference graphs. We prove that Vizing's total-color conjecture holds for dually chordal graphs. We describe a new heuristic that Fields an exact total coloring for even maximum degree dually chordal graphs and an exact edge coloring for odd maximum degree dually chordal graphs. (C) 1999 Elsevier Science B.V. All rights reserved.70314715

Decompositions For The Edge Colouring Of Reduced Indifference Graphs

The chromatic index problem - finding the minimum number of colours required for colouring the edges of a graph - is still unsolved for indifference graphs, whose vertices can be linearly ordered so that the vertices contained in the same maximal clique are consecutive in this order. We present new positive evidence for the conjecture: every non neighbourhood-overfull indifference graph can be edge coloured with maximum degree colours. Two adjacent vertices are twins if they belong to the same maximal cliques. A graph is reduced if it contains no pair of twin vertices. A graph is overfull if the total number of edges is greater than the product of the maximum degree by ⌊n/2⌋, where n is the number of vertices. We give a structural characterization for neighbourhood-overfull indifference graphs proving that a reduced indifference graph cannot be neighbourhood-overfull. We show that the chromatic index for all reduced indifference graphs is the maximum degree. We present two decomposition methods for edge colouring reduced indifference graphs with maximum degree colours. © 2002 Elsevier Science B.V. All rights reserved.29701/03/15145155Cai, L., Ellis, J.A., NP-completeness of edge-colouring some restricted graphs (1991) Discrete Appl. Math., 30, pp. 15-27De Figueiredo, C.M.H., Meidanis, J., De Mello, C.P., A linear-time algorithm for proper interval graph recognition (1995) Inform. Process. Lett., 56, pp. 179-184De Figueiredo, C.M.H., Meidanis, J., De Mello, C.P., On edge-colouring indifference graphs (1997) Theoret. Comput. Sci., 181, pp. 91-106De Figueiredo, C.M.H., Meidanis, J., De Mello, C.P., Total-chromatic number and chromatic index of dually chordal graphs (1999) Inform. Process. Lett., 70, pp. 147-152De Figueiredo, C.M.H., Meidanis, J., De Mello, C.P., Local conditions for edge-coloring (2000) J. Combin. Math. Combin. Comput., 32, pp. 79-91. , Tech. Rep., DCC 17/95, UNICAMP, 1995Gutierrez, M., Oubiña, L., Minimum proper interval graphs (1995) Discrete Math., 142, pp. 77-85Hammer, P.L., Peled, U.N., Sun, X., Difference graphs (1990) Discrete Appl. Math., 28, pp. 35-44Hedman, B., Clique graphs of time graphs (1984) J. Combin. Theory Ser. B, 37, pp. 270-278Hilton, A.J.W., Two conjectures on edge-colouring (1989) Discrete Math., 74, pp. 61-64Holyer, I., The NP-completeness of edge-coloring (1981) SIAM J. Comput., 10, pp. 718-720Misra, J., Gries, D., A constructive proof of Vizing's theorem (1992) Inform. Process. Lett., 41, pp. 131-133Roberts, F.S., On the compatibility between a graph and a simple order (1971) J. Combin. Theory Ser. B, 11, pp. 28-38Vizing, V.G., On an estimate of the chromatic class of a p-graph (1964) Diskrete Anal., 3, pp. 25-30. , (in Russian

Coloration de graphes dirigés

Les réseaux sont omniprésents dans notre vie quotidienne, que ce soit des réseaux sociaux, des réseaux de neurones ou des réseaux routiers. Pourtant, les graphes, leur pendant théorique, sont utilisés depuis des siècles pour modéliser des problèmes pratiques. Un graphe est un ensemble de sommets reliés par des arêtes. Si on considère des arêtes orientées, on parlera plutôt de digraphes. L'un des concepts les plus féconds de la théorie des graphes (appliqué aussi bien à des problèmes d'allocation de registres qu'à l'attribution de fréquences radio) est la coloration de graphes, qui consiste à attribuer des couleurs aux sommets de manière à ce que les sommets adjacents aient des couleurs distinctes. Le nombre chromatique d'un graphe est alors le nombre minimum de couleurs nécessaires. Cette thèse s'intéresse au nombre dichromatique, une métrique introduite en 1982 par Neumann-Lara comme équivalent du nombre chromatique, mais pour les digraphes. Colorer un digraphe, c'est attribuer une couleur à chacun de ses sommets de sorte qu'aucun cycle dirigé ne soit monochromatique, et le nombre dichromatique d'un digraphe est le nombre minimum de couleurs nécessaires. Des résultats récents suggèrent que cette métrique est la bonne notion de coloration dans le cas dirigé. Le but de cette thèse est d'étudier comment la structure d'un digraphe affecte son nombre dichromatique. Dans la première partie de ce travail, nous examinons comment le nombre dichromatique interagit avec d'autres métriques. Tout d'abord, nous considérons le degré, c'est-à-dire le nombre maximum de voisins d'un sommet. Dans le cas non dirigé, cela correspond au théorème de Brooks, un théorème célèbre avec de nombreuses variations et généralisations. Dans le cas des digraphes, il n'existe pas de métrique naturelle correspondant au degré maximal. Nous étudions donc comment différentes notions de degré conduisent soit à des théorèmes de type Brooks, soit à des résultats d'impossibilité. Nous étudions également l'arc-connectivité maximale, une métrique plus générale, fournissons un théorème semblable au théorème de Brooks pour cette métrique ainsi qu'un algorithme polynomial pour reconnaître les cas extrêmaux.La deuxième partie de ce manuscrit se concentre sur un analogue dirigé de la conjecture de Gyarfas-Sumner, qui essaie de caractériser les ensembles S de graphes tels que les graphes ayant un nombre chromatique suffisamment grand contiennent un graphe de S. Cette conjecture reste largement ouverte. Pour les digraphes, une conjecture correspondante a été proposée par Aboulker, Charbit et Naserasr. Nous prouvons plusieurs cas de cette conjecture, principalement en démontrant que certaines classes de digraphes ont un nombre dichromatique borné. Par exemple, nous prouvons que les graphes orientés quasi-transitifs et localement out-transitifs ont un petit nombre dichromatique. Nous caractérisons également les digraphes qui doivent apparaître dans les orientations des graphes multipartites complets avec un nombre dichromatique suffisamment grand et, ce faisant, nous découvrons un contre-exemple à la conjecture initiale d'Aboulker, Charbit et Naserasr. Nous obtenons des résultats similaires pour les digraphes sans triangle et sans chemins dirigés sur six sommets, ainsi que pour les orientations des graphes cordaux. Dans la dernière partie de cette thèse, nous abordons le problème de l'arête-coloriage d-défectueux, qui consiste à colorer les arêtes d'un multigraphe de telle sorte que, pour tout sommet, aucune couleur n'apparaisse sur plus de d de ses arêtes incidentes. Lorsque d est égal à un, cela correspond au problème de l'arête-coloration. Shannon a trouvé une borne stricte sur le nombre de couleurs nécessaires par rapport au degré maximal lorsque d est égal à un, et nous étendons ce résultat à toute valeur de d. Nous explorons également ce problème sur des graphes simples et prouvons des résultats qui étendent le théorème de Vizing à toute valeur de d.Networks are ubiquitous in our daily life, whether they are social networks, neural networks, road networks or computer networks. Yet, graphs, their theoretical pendant, have been used for centuries to model real-life problems. A graph is a set of vertices with edges connecting them. In many applications, it is useful to give edges a direction, thus obtaining a digraph (short for directed graph). One of the most fertile concepts of graph theory (applied in a wide range of practical problems, from register allocation to mobile radio frequency assignment) is graph colouring, that consists in assigning colours to vertices so that adjacent vertices get distinct colours. The chromatic number of a graph is then the minimum number of colours required. This thesis examines the dichromatic number, a metric introduced in 1982 by Neumann-Lara as a counterpart to the chromatic number for digraphs. Colouring a digraph consists in assigning a colour to each of its vertices so that no directed cycle is monochromatic, and the dichromatic number of a digraph is the minimum number of colours needed for such a colouring. Recent results suggest that this metric is the appropriate analogue for the corresponding metric on undirected graphs. The aim of this thesis is to investigate how the structure of a digraph affects its dichromatic number and to extend various results on undirected colouring to digraphs. In the first part of this work, we examine how the dichromatic number interacts with other metrics. First, we consider the degree, which is the maximum number of neighbours of a vertex. In the undirected case, this corresponds to Brooks' theorem, a celebrated theorem with multiple variations and generalizations. In the directed case, there is no natural metric corresponding to the maximum degree, so we explore how different notions of maximum directed degree lead to either Brooks-like theorems or impossibility results. We also investigate the maximum local-arc connectivity, a metric that encompasses several degree-like metrics. We demonstrate that the dichromatic number of a digraph is upper-bounded by one plus its maximum local-arc connectivity, characterize extremal digraphs, and provide a polynomial algorithm to recognize them. The second part of this manuscript focuses on a directed analogue of the Gyarfas-Sumner conjecture. The Gyarfas-Sumner conjecture tries to characterize sets S of undirected graphs such that graphs with large enough chromatic number must contain a graph of S. This conjecture is still largely open. On digraphs, a corresponding conjecture was proposed by Aboulker, Charbit, and Naserasr. We prove several subcases of this conjecture, mainly demonstrating that certain classes of digraphs have bounded dichromatic number. For instance, we prove that quasi-transitive and locally out-transitive oriented graphs have a small dichromatic number. We also characterize digraphs that must appear in orientations of complete multipartite graphs with large enough dichromatic number and, in doing so, discover a counterexample to the initial conjecture of Aboulker, Charbit, and Naserasr. We obtain similar results for digraphs with no triangle and no directed paths on six vertices, as well as for orientations of chordal graphs. In the last part of this thesis, we address the d-edge-defective-colouring problem, which involves colouring edges of a multigraph such that, for any vertex, no colour appears on more than d of its incident edges. When d equals one, this corresponds to the infamous edge-colouring problem. Shannon established a tight bound on the number of colours needed relative to the maximum degree when d equals one, and we extend this result to any value of d. We also explore this problem on simple graphs and prove results that extend Vizing's theorem to any value of d