Connectivity of friends-and-strangers graphs on random pairs
Consider two graphs $X$ and $Y$, each with $n$ vertices. The friends-and-strangers graph $\mathsf{FS}(X,Y)$ of $X$ and $Y$ is a graph with vertex set consisting of all bijections $\sigma :V(X) \mapsto V(Y)$, where two bijections $\sigma$, $\sigma'$ are adjacent if and only if they differ precisely on two adjacent vertices of $X$, and the corresponding mappings 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. Alon, Defant, and Kravitz showed that if $X$ and $Y$ are two independent random graphs in $\mathcal{G}(n,p)$, then the threshold probability guaranteeing the connectedness of $\mathsf{FS}(X,Y)$ is $p_0=n^{-1/2+o(1)}$, and suggested to investigate the general asymmetric situation, that is, $X\in \mathcal{G}(n,p_1)$ and $Y\in \mathcal{G}(n,p_2)$. In this paper, we show that if $p_1 p_2 \ge p_0^2=n^{-1+o(1)}$ and $p_1, p_2 \ge w(n) p_0$, where $w(n)\rightarrow 0$ as $n\rightarrow \infty$, then $\mathsf{FS}(X,Y)$ is connected with high probability, which extends the result on $p_1=p_2=p$, due to Alon, Defant, and Kravitz.