Given $n$-vertex simple graphs $X$ and $Y$, 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 adjacent elements of $V(X)$ whose mappings are adjacent in $Y$. We consider the setting where $X$ and $Y$ are both edge-subgraphs of $K_{r,r}$: due to a parity obstruction, $\mathsf{FS}(X,Y)$ is always disconnected in this setting. Sharpening a result of Bangachev, we show that if $X$ and $Y$ respectively have minimum degrees $\delta(X)$ and $\delta(Y)$ and they satisfy $\delta(X) + \delta(Y) \geq \lfloor 3r/2 \rfloor + 1$, then $\mathsf{FS}(X,Y)$ has exactly two connected components. This proves that the cutoff for $\mathsf{FS}(X,Y)$ to avoid isolated vertices is equal to the cutoff for $\mathsf{FS}(X,Y)$ to have exactly two connected components. We also consider a probabilistic setup in which we fix $Y$ to be $K_{r,r}$, but randomly generate $X$ by including each edge in $K_{r,r}$ independently with probability $p$. Invoking a result of Zhu, we exhibit a phase transition phenomenon with threshold function $(\log r)/r$: below the threshold, $\mathsf{FS}(X,Y)$ has more than two connected components with high probability, while above the threshold, $\mathsf{FS}(X,Y)$ has exactly two connected components with high probability. Altogether, our results settle a conjecture and completely answer two problems of Alon, Defant, and Kravitz.

翻译：给定 $n$ 个顶点的简单图 $X$ 和 $Y$，朋友与陌生人图 $\mathsf{FS}(X, Y)$ 以所有 $n!$ 个从 $V(X)$ 到 $V(Y)$ 的双射为顶点，两个双射 $\sigma, \tau$ 相邻当且仅当它们在 $V(X)$ 的两个相邻元素上取值不同，且这两个元素在 $Y$ 中的像相邻。我们考虑 $X$ 和 $Y$ 均为 $K_{r,r}$ 的边子图的情形：由于奇偶性障碍，$\mathsf{FS}(X,Y)$ 在该情形下总是不连通的。我们改进了Bangachev的一个结果，证明若 $X$ 和 $Y$ 的最小度分别为 $\delta(X)$ 和 $\delta(Y)$，且满足 $\delta(X) + \delta(Y) \geq \lfloor 3r/2 \rfloor + 1$，则 $\mathsf{FS}(X,Y)$ 恰好有两个连通分支。这证明了 $\mathsf{FS}(X,Y)$ 避免孤立顶点的阈值等于其恰好有两个连通分支的阈值。我们还考虑一个概率设定：固定 $Y = K_{r,r}$，而随机生成 $X$，即独立地以概率 $p$ 包含 $K_{r,r}$ 中的每条边。借助Zhu的一个结果，我们展示了以 $(\log r)/r$ 为阈值函数的相变现象：低于阈值时，$\mathsf{FS}(X,Y)$ 以高概率具有多于两个连通分支；高于阈值时，$\mathsf{FS}(X,Y)$ 以高概率恰好有两个连通分支。综上，我们的结果解决了一个猜想，并完全回答了Alon、Defant和Kravitz提出的两个问题。