Colour-permuting automorphisms of complete Cayley graphs
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914754217377792 |
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| author | Alimirzaei, Shirin Morris, Dave Witte |
| author_facet | Alimirzaei, Shirin Morris, Dave Witte |
| contents | Let $G$ be a (finite or infinite) group, and let $K_G = \mathrm{Cay} ( G;G \smallsetminus \{1\} )$ be the complete graph with vertex set $G$, considered as a Cayley graph of $G$. Being a Cayley graph, it has a natural edge-colouring by sets of the form $\{s, s^{-1}\}$ for $s \in G$. We prove that every colour-permuting automorphism of $K_G$ is an affine map, unless $G \cong Q_8 \times B$, where $Q_8$ is the quaternion group of order $8$, and $B$ is an abelian group, such that $b^2$ is trivial for all $b \in B$.
We also prove (without any restriction on $G$) that every colour-permuting automorphism of $K_G$ is the composition of a group automorphism and a colour-preserving graph automorphism. This was conjectured by D.P.Byrne, M.J.Donner, and T.Q.Sibley in 2013. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_09367 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Colour-permuting automorphisms of complete Cayley graphs Alimirzaei, Shirin Morris, Dave Witte Combinatorics 05C25, 05C15 Let $G$ be a (finite or infinite) group, and let $K_G = \mathrm{Cay} ( G;G \smallsetminus \{1\} )$ be the complete graph with vertex set $G$, considered as a Cayley graph of $G$. Being a Cayley graph, it has a natural edge-colouring by sets of the form $\{s, s^{-1}\}$ for $s \in G$. We prove that every colour-permuting automorphism of $K_G$ is an affine map, unless $G \cong Q_8 \times B$, where $Q_8$ is the quaternion group of order $8$, and $B$ is an abelian group, such that $b^2$ is trivial for all $b \in B$. We also prove (without any restriction on $G$) that every colour-permuting automorphism of $K_G$ is the composition of a group automorphism and a colour-preserving graph automorphism. This was conjectured by D.P.Byrne, M.J.Donner, and T.Q.Sibley in 2013. |
| title | Colour-permuting automorphisms of complete Cayley graphs |
| topic | Combinatorics 05C25, 05C15 |
| url | https://arxiv.org/abs/2404.09367 |