The generalized 3-connectivity of lexicographic product graphs

The generalized k-connectivity kappa(k)(G) of a graph G, first introduced by Hager, is a natural generalization of the concept of (vertex-)connectivity. Denote by G circle H and G square H the lexicographic product and Cartesian product of two graphs G and H, respectively. In this paper, we prove that for any two connected graphs G and H, kappa(3)(G circle H) >= kappa(3)(G)vertical bar V(H)vertical bar. We also give upper bounds for kappa(3)(G square H) and kappa 3 (G circle H). Moreover, all the bounds are sharp.