Teorema de Hajós para Coloração Ponderada

International audienceA coloração ótima dos vértices de um grafo é um dos problemas mais estudados em teoria dos grafos devido ao número de aplicações que o problema modela e à dificuldade inerente ao problema, pois determinar o número cromático de um grafo é NP-difícil. O Teorema de Hajós clássico [Hajós, 1961] mostra uma condição necessária e suficiente para que um grafo possua número cromático pelo menos k: o grafo deve possuir um subgrafo k-construtíıvel. Este, por sua vez, é obtido a partir do grafo completo de ordem k pela aplicação de um conjunto de operações bem determinadas. Neste artigo, provamos que a coloração ponderada [Guan and Zhu, 1997] admite também uma versão do Teorema de Hajós e, portanto, apresentamos uma condição necessária e suficiente para que o número cromático ponderado de um grafo seja pelo menos k, um inteiro qualquer

On the Grundy number of graphs with few P4's

International audienceThe Grundy number of a graph G is the largest number of colors used by any execution of the greedy algorithm to color G. The problem of determining the Grundy number of G is polynomial if G is a P4-free graph and NP-hard if G is a P5-free graph. In this article, we define a new class of graphs, the fat-extended P4-laden graphs, and we show a polynomial time algorithm to determine the Grundy number of any graph in this class. Our class intersects the class of P5-free graphs and strictly contains the class of P4-free graphs. More precisely, our result implies that the Grundy number can be computed in polynomial time for any graph of the following classes: P4-reducible, extended P4-reducible, P4-sparse, extended P4-sparse, P4-extendible, P4-lite, P4-tidy, P4-laden and extended P4-laden, which are all strictly contained in the fat-extended P4-laden class

Grundy number on P4-classes

International audienceIn this article, we define a new class of graphs, the fat-extended P4 -laden graphs, and we show a polynomial time algorithm to determine the Grundy number of the graphs in this class. This result implies that the Grundy number can be found in polynomial time for any graph of the following classes: P4 -reducible, extended P4 -reducible, P4 -sparse, extended P4 -sparse, P4 -extendible, P4 -lite, P4 -tidy, P4 -laden and extended P4 -laden, which are all strictly contained in the fat-extended P4 - laden class

Weighted Coloring on P4-sparse Graphs

International audienceGiven an undirected graph G = (V, E) and a weight function w : V → R+, a vertex coloring of G is a partition of V into independent sets, or color classes. The weight of a vertex coloring of G is defined as the sum of the weights of its color classes, where the weight of a color class is the weight of a heaviest vertex belonging to it. In the WEIGHTED COLORING problem, we want to determine the minimum weight among all vertex colorings of G [1]. This problem is NP-hard on general graphs, as it reduces to determining the chromatic number when all the weights are equal. In this article we study the WEIGHTED COLORING problem on P4-sparse graphs, which are defined as graphs in which every subset of five vertices induces at most one path on four vertices [2]. This class of graphs has been extensively studied in the literature during the last decade, and many hard optimization problems are known to be in P when restricted to this class. Note that cographs (that is, P4-free graphs) are P4-sparse, and that P4-sparse graphs are P5-free. The WEIGHTED COLORING problem is in P on cographs [3] and NP-hard on P5-free graphs [4]. We show that WEIGHTED COLORING can be solved in polynomial time on a subclass of P4-sparse graphs that strictly contains cographs, and we present a 2-approximation algorithm on general P4-sparse graphs. The complexity of WEIGHTED COLORING on P4- sparse graphs remains open

Proper orientation of cacti

An orientation of a graph is proper if two adjacent vertices have different indegrees. We prove that every cactus admits a proper orientation with maximum indegree at most 7. We also prove that the bound 7 is tight by showing a cactus having no proper orientation with maximum indegree less than 7. We also prove that any planar claw-free graph has a proper orientation with maximum indegree at most 6 and that this bound can also be attained

The Proportional Coloring Problem: Optimizing Buffers in Radio Mesh Networks

International audienceIn this paper, we consider a new edge coloring problem to model call scheduling op- timization issues in wireless mesh networks: the proportional coloring. It consists in finding a minimum cost edge coloring of a graph which preserves the propor- tion given by the weights associated to each of its edges. We show that deciding if a weighted graph admits a proportional coloring is pseudo-polynomial while de- termining its proportional chromatic index is NP-hard. We then give lower and upper bounds for this parameter that can be computed in pseudo-polynomial time. We finally identify a class of graphs and a class of weighted graphs for which the proportional chromatic index can be exactly determined

Proper orientation of cacti

International audienceAn orientation of a graph G is proper if two adjacent vertices have different in-degrees. The proper-orientation number − → χ (G) of a graph G is the minimum maximum in-degree of a proper orientation of G. In [1], the authors ask whether the proper orientation number of a planar graph is bounded. We prove that every cactus admits a proper orientation with maximum in-degree at most 7. We also prove that the bound 7 is tight by showing a cactus having no proper orientation with maximum in-degree less than 7. We also prove that any planar claw-free graph has a proper orientation with maximum in-degree at most 6 and that this bound can also be attained

Improper colouring of weighted grid and hexagonal graphs

International audienceWe study a weighted improper colouring problem motivated by a frequency allocation problem. It consists of associating to each vertex a set of p(v) (weight) distinct colours (frequencies), such that the set of vertices having a given colour induces a graph of degree at most k (the case k = 0 corresponds to a proper coloring). The objective is to minimize the number of colors. We propose approximation algorithms to compute such colouring for general graphs. We apply these to obtain good approximation ratio for grid and hexagonal graphs. Furthermore we give exact results for the 2-dimensional grid and the triangular lattice when the weights are all the same

Allocation de fréquences et coloration impropre des graphes hexagonaux pondérés

National audienceMotivés par un problème d'allocation de fréquences, nous étudions la coloration impropre des graphes pondérés et plus particulièrement des graphes hexagonaux pondérés. Nous donnons des algorithmes d'approximation pour trouver de telles colorations