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

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

ALGORITHMICA

SPRINGER

Lugar: Berlin; Año: 2015 vol. 73 p. 289 - 305

0178-4617

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 least 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.