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For a graph $G = (V, E)$, a subset $F\subset V(G)$ is called an$R_k$-vertex-cut of $G$ if $G -F$ is disconnected and each vertex $u \in V(G)-F$ has at least $k$ neighbors in $G -F$. The $R_k$-vertex-connectivity of $G$,denoted by $\kappa^k(G)$, is the cardinality of the minimum $R_k$-vertex-cut of$G$, which is a refined measure for the fault tolerance of network $G$. In thispaper, we study $\kappa^2$ for Cayley graphs generated by $k$-trees. Let$Sym(n)$ be the symmetric group on $\{1, 2, \cdots ,n\}$ and $\mathcal{T}$ be aset of transpositions of $Sym(n)$. Let $G(\mathcal{T})$ be the graph on $n$vertices $\{1, 2, . . . ,n\}$ such that there is an edge $ij$ in$G(\mathcal{T})$ if and only if the transposition $ij\in \mathcal{T}$. Thegraph $G(\mathcal{T})$ is called the transposition generating graph of$\mathcal{T}$. We denote by $Cay(Sym(n),\mathcal{T})$ the Cayley graphgenerated by $G(\mathcal{T})$. The Cayley graph $Cay(Sym(n),\mathcal{T})$ isdenoted by $T_kG_n$ if $G(\mathcal{T})$ is a $k$-tree. We determine$\kappa^2(T_kG_n)$ in this work. The trees are $1$-trees, and the completegraph on $n$ vertices is a $n-1$-tree. Thus, in this sense, this work is ageneralization of the such results on Cayley graphs generated by transpositiongenerating trees and the complete-transposition graphs.

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