b-Coloring is NP-Hard on Co-Bipartite Graphs and Polytime Solvable on Tree-Cographs

  • Flavia Bonomo
  • Oliver Schaudt
  • Maya Stein
  • Mario Valencia-Pabon
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8596)

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. 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. 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. 3.

    We describe a polynomial time dynamic programming algorithm to compute the b-chromatic number of co-trees.

     
  4. 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.

     

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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Flavia Bonomo
    • 1
  • Oliver Schaudt
    • 2
  • Maya Stein
    • 3
  • Mario Valencia-Pabon
    • 4
  1. 1.CONICET and Dep. de Computación, Facultad de Ciencias Exactas y NaturalesUniversidad de Buenos AiresBuenos AiresArgentina
  2. 2.Institut de Mathématiques de Jussieu, CNRS UMR7586Université Pierre et Marie Curie (Paris 6)ParisFrance
  3. 3.Centro de Mod. Mat.Universidad de ChileSantiagoChile
  4. 4.Université Paris 13, Sorbonne Paris Cité, LIPN, CNRS UMR7030VilletaneuseFrance

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