Some results on domination number of products of graphs

Abstract.Let G= (V,E) be a simple graph. A subset D of V is called a dominating set of G if for every vertex κ,εV—D,κ 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 GXH is the Cartesian product of G and H, then $$\gamma (G \times H) \geqslant \gamma (G) \cdot \gamma (H).$$ . In this paper, it is showed that the conjecture holds if Y(H)=#γc(H). And for paths Pm and Pn, a lower bound and an upper bound for γ(PmXPn) are obtained.