InforLorV4, Main, Exploration, bibRecord, 000D24

b-coloring is NP-hard on co-bipartite graphs and polytime solvable on tree-cographs

Identifieur interne : 000D24 ( Main/Exploration ); précédent : 000D23; suivant : 000D25

b-coloring is NP-hard on co-bipartite graphs and polytime solvable on tree-cographs

Auteurs : Flavia Bonomo [Argentine] ; Oliver Schaudt [France] ; Maya Stein [Chili] ; Mario Valencia-Pabon [France]

Source :

Algorithmica [ 0178-4617 ] ; 2014.

RBID : Hal:hal-01102516

Abstract

A b-coloring of a graph is a proper coloring such that every color class contains a vertex that is adjacent to all other color classes. The b-chromatic number of a graph G, denoted by \chi_b(G), is the maximum number t such that G admits a b-coloring with t colors. A graph G is called b-continuous if it admits a b-coloring with t colors, for every t = \chi(G),\ldots,\chi_b(G), and b-monotonic if \chi_b(H_1) \geq \chi_b(H_2) for every induced subgraph H_1 of G, and every induced subgraph H_2 of H_1. We investigate the b-chromatic number of graphs with stability number two. These are exactly the complements of triangle-free graphs, thus including all complements of bipartite graphs. The main results of this work are the following: - We characterize the b-colorings of a graph with stability number two in terms of matchings with no augmenting paths of length one or three. We derive that graphs with stability number two are b-continuous and b-monotonic. - We prove that it is NP-complete to decide whether the b-chromatic number of co-bipartite graph is at most a given threshold. - We describe a polynomial time dynamic programming algorithm to compute the b-chromatic number of co-trees. - Extending several previous results, we show that there is a polynomial time dynamic programming algorithm for computing the b-chromatic number of tree-cographs. Moreover, we show that tree-cographs are b-continuous and b-monotonic.

Url:

https://hal.archives-ouvertes.fr/hal-01102516

DOI: 10.1007/s00453-014-9921-5

Affiliations:

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Le document en format XML

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<front><div type="abstract" xml:lang="en">A b-coloring of a graph is a proper coloring such that every color class contains a vertex that is adjacent to all other color classes. The b-chromatic number of a graph G, denoted by \chi_b(G), is the maximum number t such that G admits a b-coloring with t colors. A graph G is called b-continuous if it admits a b-coloring with t colors, for every t = \chi(G),\ldots,\chi_b(G), and b-monotonic if \chi_b(H_1) \geq \chi_b(H_2) for every induced subgraph H_1 of G, and every induced subgraph H_2 of H_1. We investigate the b-chromatic number of graphs with stability number two. These are exactly the complements of triangle-free graphs, thus including all complements of bipartite graphs. The main results of this work are the following: - We characterize the b-colorings of a graph with stability number two in terms of matchings with no augmenting paths of length one or three. We derive that graphs with stability number two are b-continuous and b-monotonic. - We prove that it is NP-complete to decide whether the b-chromatic number of co-bipartite graph is at most a given threshold. - We describe a polynomial time dynamic programming algorithm to compute the b-chromatic number of co-trees. - Extending several previous results, we show that there is a polynomial time dynamic programming algorithm for computing the b-chromatic number of tree-cographs. Moreover, we show that tree-cographs are b-continuous and b-monotonic.</div>
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b-coloring is NP-hard on co-bipartite graphs and polytime solvable on tree-cographs

b-coloring is NP-hard on co-bipartite graphs and polytime solvable on tree-cographs

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Abstract

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Le document en format XML

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