$b$-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 $\chi _b(H_1) \ge \chi _b(H_2)$ 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 co-bipartite graph is at least 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.