On Vizing's conjecture

Abstract

A dominating set D for a graph G is a subset of V (G) such that any vertex in V (G)−D has a neighbor in D, and a domination number γ(G) is the size of a minimum dominating set for G. For the Cartesian product G2H Vizing’s conjecture [10] states that γ(G2H) ≥ γ(G)γ(H) for every pair of graphs G, H. In this paper we introduce a new concept which extends the ordinary domination of graphs, and prove that the conjecture holds when γ(G) = γ(H) = 3.