Connectivity of friends-and-strangers graphs on random pairs

Consider two graphs X and Y, each with n vertices. The friends-and-strangers graph FS(X,Y) of Y is a vertex set consisting all bijections σ:V(X)↦V(Y), where σ, σ′ are adjacent if only they differ precisely on vertices X, the corresponding mappings in Y. most fundamental question that one can ask about these whether or not connected. Recently, Alon, Defant Kravitz showed independent random G(n,p), then threshold probability guaranteeing connectedness p0=n−1/2+o(1), suggested to investigate general asymmetric situation, is, X∈G(n,p1) Y∈G(n,p2). In this paper, we show p1p2≥p02=n−1+o(1) p1,p2≥w(n)p0, w(n)→0 as n→∞, connected high probability, which extends result p1=p2=p, due Kravitz.