On the Grundy and b-Chromatic Numbers of a Graph

On the Grundy and b-Chromatic Numbers of a Graph

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Article ID: iaor20131259
Volume: 65
Issue: 4
Start Page Number: 885
End Page Number: 899
Publication Date: Apr 2013
Journal: Algorithmica
Authors: ,
Keywords: combinatorial optimization
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 bchromatic number of a graph G, denoted by χ b (G), is the largest k such that G has a bcolouring with k colours, that is a colouring in which each colour class contains a bvertex, 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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