Dominating direct products of graphs

An upper bound for the domination number of the direct product of graphs is proved. It in particular implies that for any graphs G and H  , γ(G×H)⩽3γ(G)γ(H)γ(G×H)⩽3γ(G)γ(H). Graphs with arbitrarily large domination numbers are constructed for which this bound is attained. Concerning the upper domination number we prove that Γ(G×H)⩾Γ(G)Γ(H)Γ(G×H)⩾Γ(G)Γ(H), thus confirming a conjecture from [R. Nowakowski, D.F. Rall, Associative graph products and their independence, domination and coloring numbers, Discuss. Math. Graph Theory 16 (1996) 53–79]. Finally, for paired-domination of direct products we prove that γpr(G×H)⩽γpr(G)γpr(H)γpr(G×H)⩽γpr(G)γpr(H) for arbitrary graphs G and H, and also present some infinite families of graphs that attain this bound.