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 χ 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 = χ(G), . . . , χ b (G), and b-monotonic if χ b (H 1 ) ≥ χ 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 most a given threshold.3. We describe a polynomial time dynamic programming algorithm to compute the b-chromatic number of co-trees.