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 $$\kappa _{3}(MB_{n})=n-1$$ .