Typical and extremal aspects of friends-and-strangers graphs

Given graphs X and Y with vertex sets V(X) and V(Y) of the same cardinality, the friends-and-strangers graph FS(X,Y) is the graph whose vertex set consists of all bijections σ:V(X)→V(Y), where two bijections σ and σ^′ are adjacent if they agree everywhere except for two adjacent vertices a,b∈V(X) such that σ(a) and σ(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 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 FS(X,Y) is between 3n/5+O(1) and 9n/14+O(1). When X and Y are bipartite, a parity obstruction forces FS(X,Y) to be disconnected. In this bipartite setting, we prove analogous “typical” and “extremal” results concerning when FS(X,Y) has exactly 2 connected components; for the extremal question, we obtain a nearly exact result.