On Structural Aspects of Friends-And-Strangers Graphs

Given two graphs $X$ and $Y$ with the same number of vertices, the friends-and-strangers graph $\mathsf{FS}(X, Y)$ has as its vertices all $n!$ bijections from $V(X)$ to $V(Y)$, with bijections $\sigma, \tau$ adjacent if and only if they differ on two elements of $V(X)$, whose mappings are adjacent in $Y$. In this article, we study necessary and sufficient conditions for $\mathsf{FS}(X, Y)$ to be connected for all graphs $X$ from some set. In the setting that we take $X$ to be drawn from the set of all biconnected graphs, we prove that $\mathsf{FS}(X, Y)$ is connected for all biconnected $X$ if and only if $\overline{Y}$ is a forest with trees of jointly coprime size; this resolves a conjecture of Defant and Kravitz. We also initiate and make significant progress toward determining the girth of $\mathsf{FS}(X, \text{Star}_n)$ for connected graphs $X$, and in particular focus on the necessary trajectories that the central vertex of $\text{Star}_n$ takes around all such graphs $X$ to achieve the girth.