Graph products and integer domination

We consider the {k}{k}-domination number γ{k}(G)γ{k}(G) of a graph GG and the Cartesian product G□H and the strong direct product G⊠HG⊠H of graphs GG and HH. We prove that for integers k,m≥1k,m≥1, γ{k}(G⊠H)≥γ{γ{k}(H)}(G)γ{k}(G⊠H)≥γ{γ{k}(H)}(G) and γ{km}(G⊠H)≤γ{k}(G)γ{m}(H)γ{km}(G⊠H)≤γ{k}(G)γ{m}(H), from which earlier results obtained by Fisher on γ(G⊠H)γ(G⊠H) and Fisher et al. on the fractional domination number γf(G⊠H)γf(G⊠H) were derived. We extend a result from Brešar et al. on γ(G□H) for claw-free graphs GG. We also point out some sufficient conditions for graphs to satisfy the generalized form of Vizing’s conjecture suggested by Hou and Lu.