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 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.

Partially supported by UBACyT Grant 20020100100980, PICT ANPCyT 2012-1324 and CONICET PIP 112-200901-00178 and 112-201201-00450CO (Argentina) and MathAmSud Project 13MATH-07 (Argentina–Brazil–Chile–France).

M. Valencia-Pabon – Currently Invited at INRIA Nancy - Grand Est.