The connectivity of friends-and-strangers graphs on complete multipartite graphs

TL;DR

This paper characterizes the connectivity of friends-and-strangers graphs on complete multipartite graphs, resolving a conjecture for bipartite cases and extending it to more complex structures.

Contribution

It provides a complete characterization of the connectivity of friends-and-strangers graphs on complete multipartite graphs, resolving a key conjecture and identifying conditions for exactly two components.

Findings

Connectedness characterized for complete bipartite Y

Extended results to complete multipartite Y

Identified conditions for two connected components

Abstract

For simple graphs $X$ and $Y$ on $n$ vertices, the friends-and-strangers graph $\mathsf{FS}(X,Y)$ is the graph whose vertex set consists of all bijections $\sigma: V(X) \to V(Y)$, where two bijections $\sigma$ and $\sigma'$ are adjacent if and only if they agree on all but two adjacent vertices $a, b \in V(X)$ such that $\sigma(a), \sigma(b) \in V(Y)$ are adjacent in $Y$. Resolving a conjecture of Wang, Lu, and Chen, we completely characterize the connectedness of $\mathsf{FS}(X, Y)$ when $Y$ is a complete bipartite graph. We further extend this result to when $Y$ is a complete multipartite graph. We also determine when $\mathsf{FS}(X, Y)$ has exactly two connected components where $X$ is bipartite and $Y$ is a complete bipartite graph.