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.

翻译：考虑两个$n$顶点图$X$与$Y$，其中我们将$X$解释为边代表友谊关系的社交网络，将$Y$解释为边代表相邻位置的运动图。朋友与陌生人图$\mathsf{FS}(X,Y)$是一个定义在$n!$个从$V(X)$到$V(Y)$的置换上的图，其中两个配置相邻当且仅当通过交换位于相邻位置的两个朋友，可以从其中一个配置得到另一个配置。朋友与陌生人图最初由Defant与Kravitz引入，并推广了滑块拼图问题以及令牌交换问题。先前的研究主要集中于其连通性性质。在本文中，我们研究$\mathsf{FS}(X, Y)$连通分支的直径。我们以两种方式将Kornhauser、Miller与Spirakis关于滑块拼图的结果推广至一般图。首先，我们证明当友谊图与运动图均具有较大的最小度时，$\mathsf{FS}(X, Y)$的直径具有多项式上界。其次，当两个基础图$X$与$Y$均为Erdős-Rényi随机图时，我们证明在边概率满足特定条件下，任意两个配置之间的距离几乎总是具有多项式上界。