On the Grundy and b-Chromatic Numbers of a Graph

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 .