A note on the ultimate categorical matching in a graph

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)=limn→∞ m(Gn)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×H) = m∗(G)m∗(H)?”. Actually, m∗(G×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).