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arXiv · 2303.13864

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

Leyou Xu·Bo Zhou
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Abstract

For $S\subseteq V(G)$ with $|S|\ge 2$, 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 $\kappa_k(G)=\min \{\kappa_G(S): S\in V(G) \mbox{ and } |S|=k\}$. 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 $n\ge 3$, $\kappa_4(B_n)=n-2$.

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Leyou Xu, Bo Zhou. 2023-03-24. The generalized $4$-connectivity of bubble-sort graphs. https://arxiv.org/abs/2303.13864

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