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# 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(1-3):1-16. DOI: 10.1016/S0012-365X(97)00082-4 Source: DBLP

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**ABSTRACT:**In this note, the authors generalize the ideas presented by A. Tucker in his proof of the SPGC for K 4 Gamma e-free 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 polynomial-time 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 polynomial-time algorithm for coloring perfect graphs in this new class with the size of a maximum clique colors. Key words: Perfect graphs, vertex-coloring algorithms, Tucker vertex. 1 Introduction A graph G is perfect if, for each induced subgraph H of G... -
##### Article: On Tucker vertices of graphs

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**ABSTRACT:**We give a sequential algorithm to color a new class of perfect graphs. Our algorithm is based on a 3-chromatic exchange using Tucker's algorithm for 3-coloring the perfect graphs that do not contain a clique of size four. - [Show abstract] [Hide abstract]

**ABSTRACT:**In this note, the authors generalize the ideas presented by Tucker in his proof of the Strong Perfect Graph Conjecture for (K4−e)-free graphs in order to find a vertex v in G whose special neighborhood allows to extend a ω(G)-vertex coloring of G−v to a ω(G)-vertex coloring of G. The search for such a vertex led us to the definition of p-Tucker vertices: vertices contained in at most two maximal cliques of size at least p; and to a family of classes Gp of graphs G whose maximal cliques of size at least p have no edge in common with any other maximal cliques of G. We prove that every hole-free graph G in Gp has a p-Tucker vertex and we use this fact to compute ω(G) in polynomial time for each class Gp. We state a conjecture whose validity yields the validity of the Strong Perfect Graph Conjecture for (K5−e)-free graphs clique.