The generalized 3-connectivity of Cartesian product graphsy

Abstract

The generalized connectivity of a graph, which was introduced by Chartrand et al. in 1984, is a generalization of the concept of vertex connectivity. Let S be a nonempty set of vertices of G, a collection [T 1; T 2; : : : ; T r] of trees in G is said to be internally disjoint trees connecting S if E(T i) ∩ E(T j) = ; and V (T i) ∩ V (T j) = φ for any pair of distinct integers i; j, where 1 ≤ i; j ≤ r. For an integer k with 2 ≤ k ≤ n, the k-connectivity κk(G) of G is the greatest positive integer r for which G contains at least r internally disjoint trees connecting S for any set S of k vertices of G. Obviously, κ 2 (G) = κ(G) is the connectivity of G. Sabidussi's Theorem showed that κ(G H) ≥ κ(G) + κ(H) for any two connected graphs G and H. In this paper, we prove that for any two connected graphs G and H with κ 3 (G) ≥ κ 3 (H), if κ(G) > κ 3 (G), then κ 3 (G H) ≥ κ 3 (G) + κ 3 (H); if κ(G) = κ 3 (G), then κ 3 (G H) ≥ κ 3 (G) + κ 3 (H) - 1. Our result could be seen as an extension of Sabidussi's Theorem. Moreover, all the bounds are sharp. © 2012 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France.