Two lower bounds for generalized 3-connectivity of Cartesian product graphs

The generalized k-connectivity κk(G) of a graph G, which was introduced by Chartrand et al. (1984) is a generalization of the concept of vertex connectivity. Let G and H be nontrivial connected graphs. Recently, Li et al. (2012) gave a lower bound for the generalized 3-connectivity of the Cartesian product graph G□H and proposed a conjecture for the case that H is 3-connected. In this paper, we give two different forms of lower bounds for the generalized 3-connectivity of Cartesian product graphs. The first lower bound is stronger than theirs, and the second confirms their conjecture.