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 vertices of G. In this paper, we study the Grundy number of the lexicographic, cartesian and direct products of two graphs in terms of the Grundy numbers of these graphs. Regarding the lexicographic product, we show that Γ(G) × Γ(H) ≤ Γ(G[H]) ≤ 2 Γ(G)−1 (Γ(H) − 1) + Γ(G) − 1. In addition, we show that if G is a tree or Γ(G) = ∆(G) + 1, then Γ(G[H]) = Γ(G) × Γ(H). We then deduce that for every fixed c ≤ 1, given a graph G, it is CoNP-Complete to decide if Γ(G) ≤ c × χ(G) and it is CoNP-Complete to decide if Γ(G) ≤ c × ω(G). Regarding the cartesian product, we show that there is no upper bound of Γ(G H) as a function of Γ(G) and Γ(H). Nevertheless, we prove that for any fixed graph G, there is a function h G such that, for any graph H, Γ(G H) ≤ h G (Γ(H)). Regarding the direct product, we show that Γ(G × H) ≥ Γ(G) + Γ(H) − 2 and construct for any k some graph G k such that Γ(G k) = 2k + 1 and Γ(G k × K 2) = 3k + 1.
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