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Graphs and Combinatorics

, Volume 25, Issue 2, pp 153–167 | Cite as

On the b-Coloring of Cographs and P 4-Sparse Graphs

  • Flavia Bonomo
  • Guillermo Durán
  • Frederic Maffray
  • Javier Marenco
  • Mario Valencia-Pabon
Article

Abstract

A b-coloring of a graph is a coloring such that every color class admits a vertex adjacent to at least one vertex receiving each of the colors not assigned to it. 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 b-continuous if it admits a b-coloring with t colors, for every \(t = \chi(G), \ldots, \chi_b(G)\) . We define a graph G to be 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. In this work, we prove that P 4-sparse graphs (and, in particular, cographs) are b-continuous and b-monotonic. Besides, we describe a dynamic programming algorithm to compute the b-chromatic number in polynomial time within these graph classes.

Keywords

b-coloring b-continuity b-monotonicity cographs P4-sparse graphs 

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Copyright information

© Springer Japan 2009

Authors and Affiliations

  • Flavia Bonomo
    • 1
  • Guillermo Durán
    • 2
    • 3
  • Frederic Maffray
    • 4
  • Javier Marenco
    • 5
  • Mario Valencia-Pabon
    • 6
  1. 1.CONICET and Departamento de Computación, FCENUniversidad de Buenos AiresBuenos AiresArgentina
  2. 2.CONICET and Departamento de Matemática, FCENUniversidad de Buenos AiresBuenos AiresArgentina
  3. 3.Departamento de Ingeniería Industrial, FCFMU. de ChileSantiagoChile
  4. 4.C.N.R.S., Laboratoire G-SCOPGrenobleFrance
  5. 5.Instituto de CienciasUniversidad Nacional de General SarmientoBuenos AiresArgentina
  6. 6.LIPN, Université Paris-NordVilletaneuseFrance

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