On the {k}{k}-domination number of Cartesian products of graphs

Let G□H denote the Cartesian product of graphs GG and HH. In this paper, we study the {k}{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}{k}-domination numbers of GG and HH. As applications of this lower bound, we prove that: (i) For k=1k=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}{k}-domination numbers of two any graphs GG and HH, at least one of which is a (ρ,γ)(ρ,γ)-graph, is no more than kγ{k}(G□H). (iii) The product of the {2}{2}-domination numbers of any graphs GG and HH, at least one of which is a (ρ,γ−1)(ρ,γ−1)-graph, is no more than 2γ{2}(G□H).