A partition approach to Vizing's conjecture

In 1963, Vizing [Vichysl. Sistemy 9 (1963), 30-43] conjectured that gamma(G x H) greater than or equal to gamma(G)gamma(H), where G x H denotes the cartesian product of graphs, and gamma(G) is the domination number. In this paper we define the extraction number x(G) and we prove that P-2(G) less than or equal to x(G) less than or equal to gamma(G), and gamma(G x H) greater than or equal to x(G)gamma(H), where P-2(G) is the 2-packing number of G. Though the equality x(G) = gamma(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: gamma(G x H) greater than or equal to gamma(G)P-2(H) + P-2(G)(gamma(H) - P-2(H)). (C) 1996 John Wiley & Sons, Inc.