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) boolean AND E(T-j) - empty set and V (T-i) boolean AND 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 kappa(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, kappa(2)(G) = kappa(G) is the connectivity of G. Sabidussi's Theorem showed that kappa(G square H) >= kappa(G) + kappa(H) for any two connected graphs G and H. In this paper, we prove that for any two connected graphs G and H with kappa(3) (G) >= kappa(3) (H), if kappa(G) > kappa(3) (G), then kappa(3) (G square H) >= kappa(3) (G) + kappa(3) (H); if kappa(G) = kappa(3)(G), then kappa(3)(G square H) >= kappa(3)(G) + kappa(3) (H) - 1. Our result could be seen as an extension of Sabidussi's Theorem. Moreover, all the bounds are sharp.

Source : oai:HAL:hal-00990569v1

Volume: Vol. 14 no. 1

Section: Graph Theory

Published on: March 13, 2012

Submitted on: May 4, 2011

Keywords: Connectivity,Generalized connectivity,Internally disjoint path,Internally disjoint trees,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]

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