On Two Generalized Connectivities of Graphs

Sun, Yuefang  ; Li, Fengwei ; Jin, Zemin

Discussiones Mathematicae. Graph Theory, Tome 38 (2018) no. 1, pp. 245-261

Résumé

The concept of generalized k-connectivity κ_k (G), mentioned by Hager in 1985, is a natural generalization of the path-version of the classical connectivity. The pendant tree-connectivity τ_k (G) was also introduced by Hager in 1985, which is a specialization of generalized k-connectivity but a generalization of the classical connectivity. Another generalized connectivity of a graph G, named k-connectivity κ_k^' (G), introduced by Chartrand et al. in 1984, is a generalization of the cut-version of the classical connectivity. In this paper, we get the lower and upper bounds for the difference of κ_k^' (G) and τ_k(G) by showing that for a connected graph G of order n, if κ_k^' (G) n − k + 1 where k ≥ 3, then 1 ≤κ_k^' (G) − τ_k (G) ≤ n − k; otherwise, 1 ≤ κ_k^' (G) − τ_k(G) ≤ n − k + 1. Moreover, all of these bounds are sharp. We get a sharp upper bound for the 3-connectivity of the Cartesian product of any two connected graphs with orders at least 5. Especially, the exact values for some special cases are determined. Among our results, we also study the pendant tree-connectivity of Cayley graphs on Abelian groups of small degrees and obtain the exact values for τ_k(G), where G is a cubic or 4-regular Cayley graph on Abelian groups, 3 ≤ k ≤ n.

Détail

Keywords: k -connectivity, pendant tree-connectivity, Cartesian product, Cayley graph