The Generalized Three-Connectivity of Two Kinds of Cayley Graphs

Abstract

Let $S \subseteq V (G)$ and $\kappa_{G}(S)$ denote the maximum number $r$ of edge-disjoint trees $T_1, T_2,\ldots, T_r$ in G such that $V (T_i) \cap V (T_j) =S$ for any $i,j \in \{ 1, 2, \ldots, r\}$ and $i \neq j$ . For an integer k with $2 \leq k \leq n$ , the generalized k-connectivity of a graph G is defined as $\kappa_k (G) = \min \{\kappa_{G} (S) \vert S \subseteq V (G)$ and $\vert S \vert = k\}$ . The generalized k-connectivity is a generalization of traditional connectivity. In this paper, we focus on the Cayley graph generated by complete graphs and the Cayley graph generated by wheel graphs, denoted by $CT_n$ and $WG_n$ , respectively. We study the generalized 3-connectivity of the two kinds of graphs and show that $\kappa_{3} (CT_n) = {n(n-1) \over 2} - 1$ and $\kappa_3 (WG_n) = 2n - 3$ for $n \geq 3$ .