Path-connectivity of lexicographic product graphs

Dirac showed that in a (k - 1)-connected graph there is a path through all the k vertices. The k-path-connectivity pi(k)(G) of a graph G, which is a generalization of Dirac's notion, was introduced by Hager in 1986. Denote by G. H the lexicographic product of two graphs G and H. In this paper, we prove that pi(3)(G circle H) >= pi(3)(G) left perpendicular(vertical bar V(H)vertical bar - 1)/2 right perpendicular + 1 for any two connected graphs G and H. Moreover, the bound is sharp. We also derive an upper bound of pi(3)(G circle H), that is, pi(3)(G circle H) <= 2 pi(3)(G)vertical bar V(H)vertical bar.