On Structural Aspects of Friends-And-Strangers Graphs

Abstract

Given two graphs X and Y with the same number of vertices, the friends-and-strangers graph FS(X, Y ) has as its vertices all n! bijections from V (X) to V (Y ), with bijections σ, τ adjacent if and only if they differ on two elements of V (X), whose mappings are adjacent in Y . In this article, we study necessary and sufficient conditions for FS(X, Y ) to be connected for all graphs X from some set. In the setting that we take X to be drawn from the set of all biconnected graphs, we prove that FS(X, Y ) is connected for all biconnected X if and only if Y is a forest with trees of jointly coprime size; this resolves a conjecture of Defant and Kravitz. We also initiate and make significant progress toward determining the girth of FS(X, Starn) for connected graphs X, and in particular focus on the necessary trajectories that the central vertex of Starn takes around all such graphs X to achieve the girth.

Connectedness and cycle spaces of friends-and-strangers graphs

“…Several papers have continued this line of work when one of X or Y is fixed to be a specific type of graph [5,7], when X and Y are Erdős-Rényi random graphs [1,11], or when X and Y satisfy certain minimum-degree conditions [1,3]. Jeong recently studied the girths and diameters of friendsand-strangers graphs [8]. The first author has also related friends-and-strangers graphs of the form FS(Cycle n , Y ), where Cycle n is the cycle with n vertices, to a dynamical system called toric promotion [4].…”