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

BONOMO, FLAVIA; SCHAUDT, OLIVER; STEIN, MAYA; VALENCIA-PABON, MARIO

Lisboa

Simposio; International Symposium on Combinatorial Optimization (ISCO); 2014

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),...,chi_b(G), and b-monotonic if chi_b(H_1) >= 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: 1) 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. 2) We prove that it is NP-complete to decide whether the b-chromatic number of a co-bipartite graph is at most a given threshold. 3) We describe a polynomial time dynamic programming algorithm to compute the b-chromatic number of co-trees. 4) 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.