Typical and Extremal Aspects of Friends-and-Strangers Graphs
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Given graphs X and Y with vertex sets V(X) and V(Y) of the same cardinality, the friends-and-strangers graph π₯π²(X,Y) is the graph whose vertex set consists of all bijections Ο:V(X)β V(Y), where two bijections Ο and Ο' are adjacent if they agree everywhere except for two adjacent vertices a,b β V(X) such that Ο(a) and Ο(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 ErdΕs-RΓ©nyi random graphs with n vertices and edge probability p, then the threshold probability guaranteeing the connectedness of π₯π²(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 π₯π²(X,Y) is between 3n/5+O(1) and 9n/14+O(1). When X and Y are bipartite, a parity obstruction forces π₯π²(X,Y) to be disconnected. In this bipartite setting, we prove analogous "typical" and "extremal" results concerning when π₯π²(X,Y) has exactly 2 connected components; for the extremal question, we obtain a nearly exact result.
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