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 = 1 t K n i ) ≥ 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 ) − 1 . Infinite families of graphs that attain the bound are presented. For these graphs it also holds that γ t ( G × H ) = γ ( G ) + γ ( H ) − 1 . Some additional parallels with the total domination number are made.