The generalized 3-connectivity of star graphs and bubble-sort graphs

For S ⊆ G, let K(S) denote the maximum number r of edge‐disjoint trees $T_1,T_2,\dots ,T_r$ in G such that $V\left(T_i\right)nV\left(T_j\right)=S$ for any $i,j\in \{1,2,\dots ,r\}$ and i ≠ j. For every 2 ≤ k ≤ n, the generalized k‐connectivity of G Kk (G) is defined as the minimum K(S) over all k‐subsets S of vertices, i.e., $K_k\left(G\right)=$ min $\left\{K\right(S\left)\right|S\subseteq V\left(G\right)\mathrm{and}\left|S\right|=k\}$ . Clearly, K 2(G) corresponds to the traditional connectivity of G. The generalized k‐connectivity can serve for measuring the capability of a network G to connect any k vertices in G. Cayley graphs have been used extensively to design interconnection networks. In this paper, we restrict our attention to two classes of Cayley graphs, the star graphs Sn and the bubble‐sort graphs Bn , and investigate the generalized 3‐connectivity of Sn and Bn . We show that $K_3\left(S_n\right)=n‐2$ and $K_3\left(B_n\right)=n‐2$ .