Given graphs X and Y with vertex sets $V(X)$ and $V(Y)$ of the same cardinality, the friends-and-strangers graph $\mathsf{FS}(X,Y)$ is the graph whose vertex set consists of all bijections $\sigma :V(X)\to V(Y)$, where two bijections σ and ${\sigma }^{\prime }$ are adjacent if they agree everywhere except for two adjacent vertices $a,b\in V(X)$ such that $\sigma (a)$ and $\sigma (b)$ are adjacent in Y. The most fundamental question that one can ask about these friends-and-strangers graphs is whether or not they are connected; we address this problem from two different perspectives. First, we address the case of “typical” X and Y by proving that if X and Y are independent Erdős-Rényi random graphs with n vertices and edge probability p, then the threshold probability guaranteeing the connectedness of $\mathsf{FS}(X,Y)$ with high probability is $p=n^{-1/2+o(1)}$. Second, we address the case of “extremal” X and Y by proving that the smallest minimum degree of the n-vertex graphs X and Y that guarantees the connectedness of $\mathsf{FS}(X,Y)$ is between $3n/5+O(1)$ and $9n/14+O(1)$. When X and Y are bipartite, a parity obstruction forces $\mathsf{FS}(X,Y)$ to be disconnected. In this bipartite setting, we prove analogous “typical” and “extremal” results concerning when $\mathsf{FS}(X,Y)$ has exactly 2 connected components; for the extremal question, we obtain a nearly exact result.

给定带有顶点集的图X和Y$五(X)$和$五(是)$相同基数的朋友和陌生人图$\mathsf{FS}(X,是)$是顶点集由所有双射组成的图$\sigma ：五(X)\to 五(是)$, 其中两个双射σ和${\sigma }^{\text{'}}$如果它们在除两个相邻顶点之外的所有地方都一致，则它们是相邻的$\mathrm{一种},b\in 五(X)$这样$\sigma (\mathrm{一种})$和$\sigma (b)$在Y中相邻。关于这些朋友和陌生人图，人们可以问的最基本的问题是它们是否相互关联；我们从两个不同的角度来解决这个问题。首先，我们通过证明如果X和Y是具有n个顶点和边概率p的独立 Erdős-Rényi 随机图来解决“典型” X和Y的情况，那么保证连接性的阈值概率$\mathsf{FS}(X,是)$很有可能是$p=n^{-1/2+○(1)}$. 其次，我们通过证明保证连接性的n 个顶点图X和Y的最小最小度来解决“极值” X和Y的情况$\mathsf{FS}(X,是)$在。。。之间$3n/5+○(1)$和$9n/14+○(1)$. 当X和Y是二分时，奇偶阻塞力$\mathsf{FS}(X,是)$被断开。在这种二分设置中，我们证明了关于何时的类似“典型”和“极端”结果$\mathsf{FS}(X,是)$正好有 2 个连通分量；对于极值问题，我们得到了几乎精确的结果。