Article
An algorithm for coloring some perfect graphs
University of Science and Technology Houari Boumediene, Le Retour de la Chasse, Alger, Algeria
Discrete Mathematics (Impact Factor: 0.56). 03/1998; 183(13):116. DOI: 10.1016/S0012365X(97)000824 Source: DBLP
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 "This class of graph was introduced by Ait Haddadene, Gravier and Maffray in 1998 [2].A vertex v is called MAG vertex if the union of any four triangle in N (v) contain a K 4 .A graph is called MAG graph if any induced subgraph contain a MAG vertex.This class of graph contain a chordal graphs, a (K 4 − e)free graphs, a K 4 free and line graphs of bipartite graphs. The main result in [3]is the following: Theorem(Ait Haddedene and al [2]): Let G be a MAG Berge graph and v be a MAG vertex of G. Then from any ω(G)coloring of G−v one can obtain in polynomial time an ω(G)coloring of G. The algorithmic proof of the previous result suggest a combinatorial algorithm in O(n 4 ) when the appropriate vertex ordering was given [3]. "
[Show abstract] [Hide abstract] ABSTRACT: The graph is perfect, if in all its induced subgraphs the size of the largest clique is equal to the chromatic number.In 1960 Berge formulated two conjectures about perfct graphs one stronger than the other, the weak perfect conjecture was proved in 1972 by Lovasz and the strong perfect onjecture was proved in 2003 by Chudnovsky and al.The problem to determine an optimal coloring of a graph is NPcomplete in general case.Grötschell and al developed polynomial algorithm to solve this problem for the whole of the perfect graphs. Indeed their algorithms are not practically effcient. Thus, the search of very effcient polynomial algorithms to solve these problem in the case of the perfect graphs continues to have a practical interest.We will review the wealth of results that have appeared on these topics using bichromatic, trichromatic exchange and contraction operation. 
 "It may then be interesting to find classes of graphs that always have a Tucker vertex. This question was investigated in [1, 2, 3, 22, 43] for several classes of graphs; as their definitions are quite technical, we refer the interested reader directly to these articles. "
Chapter: On the coloration of perfect graphs
[Show abstract] [Hide abstract] ABSTRACT: We consider only finite graphs, without loops. Given an undirected graph G = (V, E), a kcoloring of the vertices of G is a mapping c: V → {1, 2,..., k} for which every edge xy of G has c(x) ≠ c(y). If c(v) = i we say that v has color i. Those sets c1(T) (i = 1,..., k) that are not empty are called the color classes of the coloring c. Each color class is clearly a stable set (i.e., a subset of vertices with no edge between any two of them), hence we will frequently view a coloring as a partition into stable sets. The graph G is called kcolorable if it admits a kcoloring, and the chromatic number of G, denoted by χ(G), is the smallest integer k such that G is kcolorable. We refer to [9, 16, 29] for general results on graph theory.  [Show abstract] [Hide abstract] ABSTRACT: In this note, the authors generalize the ideas presented by A. Tucker in his proof of the SPGC for K 4 Gamma efree graphs in order to find a vertex v (called here a Tucker vertex) in G whose special neighborhood allows to extend a previous coloring of G Gamma v. The search of such a vertex led us to define a property on the intersection of large cliques and a family of classes of graphs that satisfy this property. We prove that every graph G in this family has a Tucker vertex and we use this fact to give a polynomialtime algorithm to compute !(G). We give a proof of the SPGC for a new class: graphs where every edge of a maximal clique of size at least 4 belongs to precisely that clique. The proof directly yields a combinatorial polynomialtime algorithm for coloring perfect graphs in this new class with the size of a maximum clique colors. Key words: Perfect graphs, vertexcoloring algorithms, Tucker vertex. 1 Introduction A graph G is perfect if, for each induced subgraph H of G...