Snarks with total chromatic number 5

A k-total-coloring of G is an assignment of k colors to the edges and vertices of G, so that adjacent and incident elements have different colors. The total chromatic number of G, denoted by chi(T)(G), is the least k for which G has a k-total-coloring. It was proved by Rosenfeld that the total chromatic number of a cubic graph is either 4 or 5. Cubic graphs with chi(T) = 4 are said to be Type 1, and cubic graphs with chi(T) = 5 are said to be Type 2. Snarks are cyclically 4-edge-connected cubic graphs that do not allow a 3-edge-coloring. In 2003, Cavicchioli et al. asked for a Type 2 snark with girth at least 5. As neither Type 2 cubic graphs with girth at least 5 nor Type 2 snarks are known, this is taking two steps at once, and the two requirements of being a snark and having girth at least 5 should better be treated independently. In this paper we will show that the property of being a snark can be combined with being Type 2. We will give a construction that gives Type 2 snarks for each even vertex number n >= 40. We will also give the result of a computer search showing that among all Type 2 cubic graphs on up to 32 vertices, all but three contain an induced chordless cycle of length 4. These three exceptions contain triangles. The question of the existence of a Type 2 cubic graph with girth at least 5 remains open

Minimum-Density Identifying Codes in Square Grids

International audienceAn identifying code in a graph G is a subset of vertices with the property that for each vertex v ∈ V (G), the collection of elements of C at distance at most 1 from v is non-empty and distinct from the collection of any other vertex. We consider the minimum density d * (S k) of an identifying code in the square grid S k of height k (i.e. with vertex set Z × {1,. .. , k}). Using the Discharging Method, we prove 7 20 + 1 20k ≤ d * (S k) ≤ min 2 5 , 7 20 + 3 10k , and d * (S3) = 7 18

A Charming Class of Perfectly Orderable Graphs

We investigate the following conjecture of Vašek Chvátal: any weakly triangulated graph containing no induced path on five vertices is perfectly orderable. In the process we define a new polynomially recognizable class of perfectly orderable graphs called charming. We show that every weakly triangulated graph not containing as an induced subgraph a path on five vertices or the complement of a path on six vertices is charming

Sur les colorations des arêtes des graphes cubiques

Université : Université scientifique et médicale de Grenobl

Minimisation d'une fonction sous-modulaire graphique ou le problème de la coopération optimale

International audienc

Coloration d'hypergraphes et clique-coloration

Le travail de cette thèse s'est porté sur certains problèmes de coloration d'hypergraphes, dont certains sont en lien avec les graphes parfaits. Dans un premier temps, la coloration des hypergraphes est abordée de manière générale, et nous y démontrons une conjecture de Sterboul (généralisant un précédent résultat de Fournier et Las Vergnas) affirmant que si un hypergraphe ne contient pas un type particulier de cycle impair, alors il est 2-coloriable. Par la suite nous étudions plus précisément le problème de clique-coloration : une clique maximale d'un graphe est un sous-graphe complet, maximal par inclusion. Le problème consiste a colorier les sommets du graphe de sorte que chaque clique maximale contienne aux moins deux sommets de couleurs distinctes. Le point de départ de cette thèse était de savoir s'il existe une constante k telle que tous les graphes parfaits sont k-clique-coloriables. Cette question n'est toujours pas résolue, bien qu'on ne connaisse aucun graphe sans trou impair qui n'est pas 3-clique-coloriable. Cependant, une telle constante existe dans de nombreux cas particuliers, dont certains (tels que les graphes sans diamant ou sans taureau) sont étudiés ici. La complexité du problème de clique-coloration est également abordée, en essayant de déterminer la la classe de complexité exacte selon les cas particuliers. Plusieurs résultats sont établis, concernant notamment la difficulté de décider si un graphe parfait est 2-clique-coloriable : ce problème est Sigma_2 P-complet, et est NP-complet pour les graphes parfaits sans K_4.GRENOBLE1-BU Sciences (384212103) / SudocSudocFranceF

Graphic Submodular Function Minimization: A Graphic Approach and Applications

International audienc

On the complexity of colouring antiprismatic graphs

International audienceA graph G is prismatic if for every triangle T of G, every vertex of G not in T has a unique neighbour in T. The complement of a prismatic graph is called \emph{antiprismatic}. The complexity of colouring antiprismatic graphs is still unknown. Equivalently, the complexity of the clique cover problem in prismatic graphs is not known. Chudnovsky and Seymour gave a full structural description of prismatic graphs. They showed that the class can be divided into two subclasses: the orientable prismatic graphs, and the non-orientable prismatic graphs. We give a polynomial time algorithm that solves the clique cover problem in every non-orientable prismatic graph. It relies on the the structural description and on later work of Javadi and Hajebi. We give a polynomial time algorithm which solves the vertex-disjoint triangles problem for every prismatic graph. It does not rely on the structural description