Diameter Bounds for Friends-and-Strangers Graphs
Amogh Akella, Rupert Li
TL;DR
This work studies the diameter of connected components in the friends-and-strangers graph FS$(X,Y)$, defining a rich combinatorial framework that generalizes sliding puzzles and token swapping. It proves polynomial-diameter bounds under strong minimum-degree conditions on $X$ and $Y$, and establishes that when $X$ and $Y$ are independent Erdős–Rényi graphs with $pq \ge C\log n / n$, the distance between any two fixed configurations is whp polynomial, specifically $O(n^5)$. The approach blends reductions to subproblems like FS$( extsf{Star}_n,\cdot)$ and FS$(K_n,\cdot)$, Wilsonian graph concepts, and probabilistic biconnectivity arguments to control local exchanges and compose global paths. Together, these results extend known sliding-puzzle diameter bounds to broad structural and random-graph regimes, while leaving open questions about the diameter in the connected case and sharper random-graph thresholds. The findings provide a framework for understanding when reachable configurations can be navigated efficiently in large, structured permutation spaces.
Abstract
Consider two $n$-vertex graphs $X$ and $Y$, where we interpret $X$ as a social network with edges representing friendships and $Y$ as a movement graph with edges representing adjacent positions. The friends-and-strangers graph $\mathsf{FS}(X,Y)$ is a graph on the $n!$ permutations $V(X)\to V(Y)$, where two configurations are adjacent if and only if one can be obtained from the other by swapping two friends located on adjacent positions. Friends-and-strangers graphs were first introduced by Defant and Kravitz, and generalize sliding puzzles as well as token swapping problems. Previous work has largely focused on their connectivity properties. In this paper, we study the diameter of the connected components of $\mathsf{FS}(X, Y)$. We extend the result of Kornhauser, Miller, and Spirakis on sliding puzzles to general graphs in two ways. First, we show that the diameter of $\mathsf{FS}(X, Y)$ is polynomially bounded when both the friendship and the movement graphs have large minimum degree. Second, when both the underlying graphs $X$ and $Y$ are Erdős-Rényi random graphs, we show that the distance between any pair of configurations is almost always polynomially bounded under certain conditions on the edge probabilities.