On Structural Aspects of Friends-And-Strangers Graphs

TL;DR

This paper characterizes when the friends-and-strangers graph is connected for all biconnected graphs X, showing it depends on the structure of Y's complement, and explores the girth of these graphs with star graphs.

Contribution

It proves a necessary and sufficient condition for the connectivity of friends-and-strangers graphs for all biconnected X, resolving a conjecture, and advances understanding of girth in these graphs.

Findings

Connectivity characterized by Y's complement being a forest with coprime-sized trees.

Resolved a conjecture of Defant and Kravitz on graph connectivity.

Progress on determining the girth of friends-and-strangers graphs with star graphs.

Abstract

Given two graphs $X$ and $Y$ with the same number of vertices, the friends-and-strangers graph $\mathsf{FS}(X, Y)$ has as its vertices all $n!$ bijections from $V(X)$ to $V(Y)$, with bijections $\sigma, \tau$ 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 $\mathsf{FS}(X, Y)$ 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 $\mathsf{FS}(X, Y)$ is connected for all biconnected $X$ if and only if $\overline{Y}$ 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 $\mathsf{FS}(X, \text{Star}_n)$ for connected graphs $X$, and in particular focus on the necessary trajectories that the central vertex of $\text{Star}_n$ takes around all such graphs $X$ to achieve the girth.