The connectivity of friends-and-strangers graphs on complete multipartite graphs
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866912204172820480 |
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| author | Zhu, Honglin |
| author_facet | Zhu, Honglin |
| contents | 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 $σ: V(X) \to V(Y)$, where two bijections $σ$ and $σ'$ are adjacent if and only if they agree on all but two adjacent vertices $a, b \in V(X)$ such that $σ(a), σ(b) \in V(Y)$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_08121 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The connectivity of friends-and-strangers graphs on complete multipartite graphs Zhu, Honglin Combinatorics 05C35 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 $σ: V(X) \to V(Y)$, where two bijections $σ$ and $σ'$ are adjacent if and only if they agree on all but two adjacent vertices $a, b \in V(X)$ such that $σ(a), σ(b) \in V(Y)$ 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. |
| title | The connectivity of friends-and-strangers graphs on complete multipartite graphs |
| topic | Combinatorics 05C35 |
| url | https://arxiv.org/abs/2307.08121 |