Lower bounds for the domination number and the total domination number of direct product graphs

A sharp lower bound for the domination number and the total domination number of the direct product of finitely many complete graphs is given: γ(×i=1tKni)≥t+1,t≥3. Sharpness is established in the case when the factors are large enough in comparison to the number of factors. The main result gives a lower bound for the domination (and the total domination) number of the direct product of two arbitrary graphs: γ(G×H)≥γ(G)+γ(H)−1γ(G×H)≥γ(G)+γ(H)−1. Infinite families of graphs that attain the bound are presented. For these graphs it also holds that γt(G×H)=γ(G)+γ(H)−1γt(G×H)=γ(G)+γ(H)−1. Some additional parallels with the total domination number are made.