The generalized $4$-connectivity of bubble-sort graphs

For $S\subseteq V(G)$ with $\mathrm{\mid }S\mathrm{\mid }\ge 2$, let ${\kappa }_G(S)$ denote the maximum number of internally disjoint trees connecting $S$ in $G$. For $2\le k\le n$, the generalized $k$-connectivity ${\kappa }_k(G)$ of an $n$-vertex connected graph $G$ is defined to be ${\kappa }_k(G)=\min \{{\kappa }_G(S):S\in V(G)and\mathrm{\mid }S\mathrm{\mid }=k\}$. The generalized $k$-connectivity can serve for measuring the fault tolerance of an interconnection network. The bubble-sort graph $B_n$ for $n\ge 2$ is a Cayley graph over the symmetric group of permutations on $[n]$ generated by transpositions from the set $\{[1,2],[2,3],\dots ,[n-1,n]\}$. In this paper, we show that for the bubble-sort graphs $B_n$ with $n\ge 3$, ${\kappa }_4(B_n)=n-2$.