### The generalized 3-connectivity of Cartesian product

*Hengzhe Li, Xueliang Li, Yuefang Sun*

#### 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})=S for any pair of distinct integers i,j, where 1≤i,j≤r. For an integer k with 2≤k≤n, the k-connectivity κ_{k}(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.Full Text: PDF PostScript