On the Grundy Number of a Graph

The Grundy number of a graph G, denoted by Gamma(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 vertices of G. Trivially Gamma(G) <= Delta(G) + 1. In this paper, we show that deciding if Gamma(G) <= Delta(G) is NP-complete. We then show that deciding if Gamma(G) >= vertical bar V(G)vertical bar - k is fixed parameter tractable with respect to the parameter k.