Grundy number and products of graphs

Abstract: The Grundy number of a graph , denoted by , is the largest such that has a greedy -colouring, that is a colouring with colours obtained by applying the greedy algorithm according to some ordering of the vertices of . In this paper, we study the Grundy number of the lexicographic and cartesian products of two graphs in terms of the Grundy numbers of these graphs. Regarding the lexicographic product, we show that . In addition, we show that if is a tree or , then . We then deduce that for every fixed , given a graph , it is CoNP-Complete to decide if and it is CoNP-Complete to decide if . Regarding the cartesian product, we show that there is no upper bound of as a function of and . Nevertheless, we prove that . [Copyright &y& Elsevier]