$$b$$b-Coloring is NP-hard on Co-bipartite Graphs and Polytime Solvable on Tree-Cographs

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 $$G$$G, denoted by $$\chi _b(G)$$χb(G), is the maximum number $$t$$t such that $$G$$G admits a b-coloring with $$t$$t colors. A graph $$G$$G is called b-continuous if it admits a b-coloring with $$t$$t colors, for every $$t = \chi (G),\ldots ,\chi _b(G)$$t=χ(G),…,χb(G), and b-monotonic if $$\chi _b(H_1) \ge \chi _b(H_2)$$χb(H1)≥χb(H2) for every induced subgraph $$H_1$$H1 of $$G$$G, and every induced subgraph $$H_2$$H2 of $$H_1$$H1. 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.