Some results on domination number of products of graphs

Let G=(V,E) be a simple graph. A subset D of V is called a dominating set of G if for every vertex x ϵ V-D, x is adjacent to at least one vertex of D. Let γ (G) and γc (G) denote the domination and connected domination number of G, respectively. In 1965, Vizing conjectured that if GΧ H is the Cartesian product of G and H, then γ(G X H)≥ γ(G) • γ (H). In this paper, it is showed that the conjecture holds if γ(H)≠ γc.(H). And for paths Pm and Pn, a lower bound and an upper bound for γ (Pm × Pn) are obtained. © 1998, Springer Verlag. All rights reserved.