A partition approach to Vizing's conjecture

In 1963, Vizing [Vichysl.Sistemy9 (1963), 30–43] conjectured that γ(G × H) ≥ γ(G)γ(H), where G × H denotes the cartesian product of graphs, and γ(G) is the domination number. In this paper we define the extraction number x(G) and we prove that P2(G) ≤ x(G), and γ(G × H) ≥ x(G)γ(H), where P2(G) is the 2-packing number of G. Though the equality x(G) = γ(G) is proven to hold in several classes of graphs, we construct an infinite family of graphs which do not satisfy this condition. Also, we show the following lower bound: γ(G × H) ≥ γ(G)P2(H) + P2(G)(γ(H) − P2(H)). © 1996 John Wiley & Sons, Inc.