Discrete Mathematics & Theoretical Computer Science, Vol 14, No 1 (2012)

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The generalized 3-connectivity of Cartesian product

Hengzhe Li, Xueliang Li, Yuefang Sun


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 {T1,T2,…,Tr} of trees in G is said to be internally disjoint trees connecting S if E(Ti)∩ E(Tj)=∅ and V(Ti)∩V(Tj)=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 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, κ2(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 κ3(G)≥κ3(H), if κ(G)>κ3(G), then κ3(G ☐ H)≥ κ3(G)+κ3(H); if κ(G)=κ3(G), then κ3(G ☐ H)≥κ3(G)+κ3(H)-1. Our result could be seen as an extension of Sabidussi's Theorem. Moreover, all the bounds are sharp.

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