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

Let G square 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 gamma({k}) (G square 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 (rho, gamma)-graph, is no more than k gamma({K}) (G square H). (iii) The product of the {2}-domination numbers of any graphs G and H, at least one of which is a (rho, gamma - 1)-graph, is no more than 2 gamma({2}) (G square H). (C) 2008 Elsevier B.V. All rights reserved.