A note on the ultimate categorical matching in a graph

Abstract

Let m(G) denote the number of vertices covered by a maximum matching in a graph G. The ultimate categorical matching m*(G) is defined as m*(G) = lim(n-->infinity)m(G(n))(1/n) where the categorical graph product is used. In (Discrete Math. 232 (2001) 1), Albert et al. ask that "Is there a graph G, with at least one edge, such that for all graphs H, m* (G x H) = m * (G)m * (H)?". Actually, m*(G x H) = m*(G)m*(H) holds for any graphs G and H with the previous result of Hsu et al. (Discrete Math. 65 (1987) 53). (C) 2002 Elsevier Science B.V. All rights reserved.