The Generalized 3-Connectivity of Cayley Graphs on Symmetric Groups Generated by Trees and Cycles

Abstract

The generalized connectivity of a graph is a natural generalization of the connectivity and can serve for measuring the capability of a network G to connect any k vertices in G. Given a graph $G=(V,E)$ and a subset $S\subseteq V$ of at least two vertices, we denote by $\kappa _G(S)$ 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 pair of distinct integers i, j, where $1\le i,j\le r$. For an integer k with $2\le k\le n$, the generalized k-connectivity is defined as $\kappa _k(G)=\min \{\kappa _G(S)| S\subseteq V(G)\ \mathrm{and}\ |S|=k\}$. That is, $\kappa _k(G)$ is the minimum value of $\kappa _G(S)$ over all k-subsets S of vertices. The study of Cayley graphs has many applications in the field of design and analysis of interconnection networks. Let Sym(n) be the group of all permutations on $\{1,\ldots ,n\}$ and ${\mathcal {T}}$ be a set of transpositions of Sym(n). Let $G({\mathcal {T}})$ be the graph on n vertices $\{1,2,\ldots ,n\}$ such that there is an edge ij in $G({\mathcal {T}})$ if and only if the transposition $[ij]\in {\mathcal {T}}$. If $G({\mathcal {T}})$ is a tree, we use the notation ${\mathbb {T}}_n$ to denote the Cayley graph $Cay(Sym(n),{\mathcal {T}})$ on symmetric groups generated by $G({\mathcal {T}})$. If $G({\mathcal {T}})$ is a cycle, we use the notation $MB_{n}$ to denote the Cayley graph $Cay(Sym(n),{\mathcal {T}})$ on symmetric groups generated by $G({\mathcal {T}})$. In this paper, we investigate the generalized 3-connectivity of ${\mathbb {T}}_{n}$ and $MB_{n}$ and show that $\kappa _{3}({\mathbb {T}}_{n})=n-2$ and $\kappa _{3}(MB_{n})=n-1$.