Local connectivity of friends-and-strangers graphs with a star

Given two finite simple graphs $X$ and $Y$ with the same number of vertices, the vertex set of the friends-and-strangers graph $\FS(X,Y)$ consists of all bijections from the vertices of $X$ to the vertices of $Y$. Two bijections are adjacent exactly when one is obtained from the other by swapping the images of two adjacent vertices of $X$, while leaving all other images unchanged, and the two swapped images are adjacent in $Y$. For $n\ge2$, let $\Star_n$ be the star on $n$ vertices. Krishnan and Li [Electron. J. Combin. 33 (2026), no. 3, Paper P3.41] conjectured that, for $n\ge3$, whenever $\FS(X,\Star_n)$ is connected, the maximum number of pairwise internally vertex-disjoint paths between any two distinct vertices is equal to the smaller of their degrees. We prove this conjecture. More precisely, if $\FS(X,\Star_n)$ is connected, then $X$ must be $2$-connected, and the conjectured equality holds for every pair of distinct vertices. We further prove the stronger result that, whenever $X$ is $2$-connected, the same equality holds in every connected component of $\FS(X,\Star_n)$.