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arXiv · 2404.09367

Colour-permuting automorphisms of complete Cayley graphs

Shirin Alimirzaei·Dave Witte Morris
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Abstract

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.

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Shirin Alimirzaei, Dave Witte Morris. 2024-04-14. Colour-permuting automorphisms of complete Cayley graphs. https://arxiv.org/abs/2404.09367

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