Grundy number and products of graphs

The Grundy number   of a graph GG, denoted by Γ(G)Γ(G), is the largest kk such that GG has a greedy  kk-colouring, that is a colouring with kk colours obtained by applying the greedy algorithm according to some ordering of the vertices of GG. 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 Γ(G)×Γ(H)≤Γ(G[H])≤2Γ(G)−1(Γ(H)−1)+Γ(G)Γ(G)×Γ(H)≤Γ(G[H])≤2Γ(G)−1(Γ(H)−1)+Γ(G). In addition, we show that if GG is a tree or Γ(G)=Δ(G)+1Γ(G)=Δ(G)+1, then Γ(G[H])=Γ(G)×Γ(H)Γ(G[H])=Γ(G)×Γ(H). We then deduce that for every fixed c≥1c≥1, given a graph GG, it is CoNP-Complete to decide if Γ(G)≤c×χ(G)Γ(G)≤c×χ(G) and it is CoNP-Complete to decide if Γ(G)≤c×ω(G)Γ(G)≤c×ω(G).Regarding the cartesian product, we show that there is no upper bound of Γ(G□H)Γ(G□H) as a function of Γ(G)Γ(G) and Γ(H)Γ(H). Nevertheless, we prove that Γ(G□H)≤Δ(G)⋅2Γ(H)−1+Γ(H)Γ(G□H)≤Δ(G)⋅2Γ(H)−1+Γ(H).