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On the Grundy and b-Chromatic Numbers of a Graph

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Abstract

The Grundy number of a graph G, denoted by Γ(G), is the largest k such that G has a greedy k-colouring, that is a colouring with k colours obtained by applying the greedy algorithm according to some ordering of the vertexes of G. The b-chromatic number of a graph G, denoted by χ b (G), is the largest k such that G has a b-colouring with k colours, that is a colouring in which each colour class contains a b-vertex, a vertex with neighbours in all other colour classes. Trivially χ b (G),Γ(G)≤Δ(G)+1. In this paper, we show that deciding if Γ(G)≤Δ(G) is NP-complete even for a bipartite graph G. We then show that deciding if Γ(G)≥|V(G)|−k or if χ b (G)≥|V(G)|−k are fixed parameter tractable problems with respect to the parameter k.

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Authors and Affiliations

  1. Projet Mascotte, I3S (CNRS, UNS) and INRIA, 2004 route des lucioles, BP 93, 06902, Sophia-Antipolis Cedex, France

    Frédéric Havet & Leonardo Sampaio

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  1. Frédéric Havet
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  2. Leonardo Sampaio
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Correspondence to Leonardo Sampaio.

Additional information

F. Havet was partially supported by ANR Blanc AGAPE, ANR-09-BLAN-0159.

L. Sampaio was partially supported by CAPES/Brazil and ANR Blanc AGAPE, ANR-09-BLAN-0159.

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Havet, F., Sampaio, L. On the Grundy and b-Chromatic Numbers of a Graph. Algorithmica 65, 885–899 (2013). https://doi.org/10.1007/s00453-011-9604-4

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  • DOI: https://doi.org/10.1007/s00453-011-9604-4

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