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For $S\subseteq V(G)$ with $|S|\ge 2$, let $\kappa_G (S)$ denote the maximumnumber 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 $\kappa_k(G)=\min \{\kappa_G(S): S\in V(G) \mbox{ and }|S|=k\}$. The generalized $k$-connectivity can serve for measuring the faulttolerance of an interconnection network. The bubble-sort graph $B_n$ for $n\ge2$ is a Cayley graph over the symmetric group of permutations on $[n]$generated by transpositions from the set $\{[1,2],[2,3],\dots, [n-1,n]\}$. Inthis paper, we show that for the bubble-sort graphs $B_n$ with $n\ge 3$,$\kappa_4(B_n)=n-2$.

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