On the path edge-connectivity of graphs

Dirac showed that in a (k - 1)-connected graph there is a path through each k vertices. The path k-connectivity pi(k)(G) of a graph G, which is a generalization of Dirac's notion, was introduced by Hager in 1986. Recently, Mao introduced the concept of path k-edge-connectivity omega(k)(G) of a graph G. Denote by G o H the lexicographic product of two graphs G and H. In this paper, we prove that omega(4)(G o H) >= omega(4)(G)[3 vertical bar V(H)vertical bar/5) I for any two graphs G and H. Moreover, the bound is sharp.