On the -domination number of Cartesian products of graphs

Let G □ H denote the Cartesian product of graphs G and H . In this paper, we study the { k } -domination number of Cartesian product of graphs and give a new lower bound of γ { k } ( G □ H ) in terms of packing and { k } -domination numbers of G and H . As applications of this lower bound, we prove that: (i) For k = 1 , the new lower bound improves the bound given by Chen, et al. [G. Chen, W. Piotrowski, W. Shreve, A partition approach to Vizing’s conjecture, J. Graph Theory 21 (1996) 103–111]. (ii) The product of the { k } -domination numbers of two any graphs G and H , at least one of which is a ( ρ , γ ) -graph, is no more than k γ { k } ( G □ H ) . (iii) The product of the { 2 } -domination numbers of any graphs G and H , at least one of which is a ( ρ , γ − 1 ) -graph, is no more than 2 γ { 2 } ( G □ H ) .