A kind of conditional connectivity of transposition networks generated by k-trees

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 κ (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 κ(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 such results on Cayley graphs generated by transposition generating trees[26] and the complete-transposition graphs[22].