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

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.
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https://hal.archives-ouvertes.fr/hal-00926924
Contributor : Mario Valencia <>
Submitted on : Friday, January 10, 2014 - 3:11:08 PM
Last modification on : Thursday, September 5, 2019 - 4:46:59 PM

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  • HAL Id : hal-00926924, version 1
  • ARXIV : 1310.8313

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Flavia Bonomo, Oliver Schaudt, Maya Stein, Mario Valencia-Pabon. b-coloring is NP-hard on co-bipartite graphs and polytime solvable on tree-cographs. 2013. ⟨hal-00926924⟩

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