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arXiv · 2307.08121

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

Honglin Zhu
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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.

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Honglin Zhu. 2023-07-16. The connectivity of friends-and-strangers graphs on complete multipartite graphs. https://doi.org/10.1007/s00026-024-00740-z

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