A kind of conditional connectivity of transposition networks generated by [formula omitted]-trees

Abstract

For a graph G=(V,E), a subset F⊂V(G) is called an Rk-vertex-cut of G if G−F is disconnected and each vertex u∈V(G)−F has at least k neighbors in G−F. The Rk-vertex-connectivity of G, denoted by κk(G), is the cardinality of the minimum Rk-vertex-cut of G, which is a refined measure for the fault tolerance of network G. In this paper, we study κ2 for Cayley graphs generated by k-trees. Let Sym(n) be the symmetric group on {1,2,…,n} and T be a set of transpositions of Sym(n). Let G(T) be the graph on n vertices {1,2,...,n} such that there is an edge ij in G(T) if and only if the transposition ij∈T. The graph G(T) is called the transposition generating graph of T. We denote by Cay(Sym(n),T) the Cayley graph generated by G(T). The Cayley graph Cay(Sym(n),T) is denoted by TkGn if G(T) is a k-tree. We determine κ2(TkGn) in this work. The trees are 1-trees, and the complete graph on n vertices is a n−1-tree. Thus, in this sense, this work is a generalization of the results on Cayley graphs generated by transposition generating trees (Yang et al., 2010) and the complete-transposition graphs (Wang et al., 2015).