Séminaire Lotharingien de Combinatoire, 85B.21 (2021), 12 pp.

Noga Alon, Colin Defant, and Noah Kravitz

Friends and Strangers Walking on Graphs

Abstract. We introduce the friends-and-strangers graph FS(X,Y) associated with graphs X and Y whose vertex sets V(X) and V(Y) have the same cardinality. This 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,bV(X) such that σ(a) and σ(b) are adjacent in Y. This setup, which has a natural interpretation in terms of friends and strangers walking on graphs, provides a common generalization of Cayley graphs of symmetric groups generated by transpositions, the famous 15-puzzle, generalizations of the 15-puzzle as studied by Wilson, and work of Stanley related to flag h-vectors. The most fundamental questions that one can ask about these friends-and-strangers graphs concern their connected components and, in particular, when there is only a single connected component.

When X is a path graph, we show that the connected components of FS(X,Y) correspond to the acyclic orientations of the complement of Y. When X is a cycle, we obtain a full description of the connected components of FS(X,Y) in terms of toric acyclic orientations of the complement of Y. In a more probabilistic vein, 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). We also study 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). Furthermore, we obtain bipartite analogues of the latter two results.

Received: December 1, 2020. Accepted: March 1, 2021. Final version: April 29, 2021.

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