On the Diameters of Friends-and-Strangers Graphs

Given simple graphs $X$ and $Y$ on the same number of vertices, the friends-and-strangers graph $\mathsf{FS}(X, Y)$ has as its vertices all bijections from $V(X)$ to $V(Y)$, where two bijections are adjacent if and only if they differ on two adjacent elements of $V(X)$ with images adjacent in $Y$. We study the diameters of connected components of friends-and-strangers graphs: the diameter of a component of $\mathsf{FS}(X,Y)$ corresponds to the largest number of swaps necessary to go from one configuration in the component to another. We show that any component of $\mathsf{FS}(\mathsf{Path}_n, Y)$ has $O(n^2)$ diameter and that any component of $\mathsf{FS}(\mathsf{Cycle}_n, Y)$ has $O(n^4)$ diameter, improvable to $O(n^3)$ whenever $\mathsf{FS}(\mathsf{Cycle}_n, Y)$ is connected. These results address an open problem posed by Defant and Kravitz. Using an explicit construction, we show that there exist $n$-vertex graphs $X$ and $Y$ such that $\mathsf{FS}(X,Y)$ has a component with $e^{\Omega(n)}$ diameter. This answers a question raised by Alon, Defant, and Kravitz in the negative. As a corollary, we observe that for such $X$ and $Y$, the lazy random walk on this component of $\mathsf{FS}(X,Y)$ has $e^{\Omega(n)}$ mixing time. This result deviates from related classical theorems regarding rapidly mixing Markov chains and makes progress on another open problem of Alon, Defant, and Kravitz. We conclude with several suggestions for future research.