Connectedness of friends-and-strangers graphs of complete bipartite graphs and others

TL;DR

This paper investigates the connectedness of friends-and-strangers graphs formed from complete bipartite graphs and arbitrary graphs, extending previous algebraic results to combinatorial methods for all bipartite sizes.

Contribution

It provides a combinatorial analysis of the connectedness of friends-and-strangers graphs for complete bipartite graphs of any size, generalizing earlier algebraic results.

Findings

Connectedness characterized for all bipartite sizes $k \, \geq 2$

Includes analysis for random graphs as $Y$

Poses open problems for future research

Abstract

Let $X$ and $Y$ be any two graphs of order $n$. The friends-and-strangers graph $\mathsf{FS}(X,Y)$ of $X$ and $Y$ is a graph with vertex set consisting of all bijections $\sigma :V(X) \mapsto V(Y)$, in which two bijections $\sigma$, $\sigma'$ are adjacent if and only if they differ precisely on two adjacent vertices of $X$, and the corresponding mappings 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. Let $K_{k,n-k}$ be a complete bipartite graph of order $n$. In 1974, Wilson characterized the connectedness of $\mathsf{FS}(K_{1,n-1},Y)$ by using algebraic methods. In this paper, by using combinatorial methods, we investigate the connectedness of $\mathsf{FS}(K_{k,n-k},Y)$ for any $Y$ and all $k\ge 2$, including $Y$ being a random graph, as suggested by Defant and Kravitz, and pose some open problems.