\(b\)-Coloring is NP-hard on Co-bipartite Graphs and Polytime Solvable on Tree-Cographs
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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.
Keywordsb-Coloring Stability number two Co-triangle-free graphs NP-hardness Treecographs Polytime dynamic programming algorithms
We would like to thank the anonymous referees for their careful reading and suggestions that helped us to improve the paper. This work was partially supported by UBACyT Grant 20020100100980, CONICET PIP 112-200901-00178 and 11220120100450CO, and ANPCyT PICT 2012–1324 (Argentina) and MathAmSud Project 13MATH-07 (Argentina–Brazil–Chile–France).
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