A Class of Graphs Approaching Vizing's Conjecture

For any graph G=(V,E), a subset S of V dominates G if all vertices are contained in the closed neighborhood of S, that is N[S]=V. The minimum cardinality over all such S is called the domination number, written γ(G). In 1963, V.G. Vizing conjectured that γ(G □ H) ≥ γ(G)γ(H) where □ stands for the Cartesian product of graphs. In this note, we define classes of graphs An, for n≥0, so that every graph belongs to some such class, and A0 corresponds to class A of Bartsalkin and German. We prove that for any graph G in class A1, γ(G□H)≥ [γ(G)-√(γ(G))]γ(H).