An Inequality Related to Vizing’s Conjecture

Let γ(G) denote the domination number of a graph G and let G H denote the Cartesian product of graphs G and H. We prove that γ(G)γ(H) ≤ 2γ(G H) for all simple graphs G and H. 2000 Mathematics Subject Classifications: Primary 05C69, Secondary 05C35 We use V (G), E(G), γ(G), respectively, to denote the vertex set, edge set and domination number of the (simple) graph G. For a pair of graphs G and H, the Cartesian product G H of G and H is the graph with vertex set V (G)× V (H) and where two vertices are adjacent if and only if they are equal in one coordinate and adjacent in the other. In 1963, V. G. Vizing [2] conjectured that for any graphs G and H, γ(G)γ(H) ≤ γ(G H). (1) The reader is referred to Hartnell and Rall [1] for a summary of recent progress on Vizing’s conjecture. We note that there are graphs G and H for which equality holds in (1). However, it was previously unknown [1] whether there exists a constant c such that γ(G)γ(H) ≤ c γ(G H). We shall show in this note that γ(G)γ(H) ≤ 2 γ(G H). For S ⊆ V (G) we let NG[S] denote the set of vertices in V (G) that are in S or adjacent to a vertex in S, i.e., the set of vertices in V (G) dominated by vertices in S.