完全凯莱图的颜色置换自同构
Colour-permuting automorphisms of complete Cayley graphs
AI总结：
研究完全凯莱图$K_G$的颜色置换自同构，证明除非特定情况其颜色置换自同构是仿射映射，且对任意$G$，其颜色置换自同构是群自同构与保色图自同构的复合，验证了前人猜想。
摘要：
设$G$为（有限或无限）群，$K_G=\mathrm{Cay}(G;G\setminus\{1\})$为以$G$为顶点集的完全图，视为$G$的凯莱图。作为凯莱图，它有自然的边着色，形式为$\{s,s^{-1}\}$，其中$s\in G$。我们证明，除非$G\cong Q_8\times B$（其中$Q_8$是8阶四元数群，$B$是阿贝尔群且对所有$b\in B$，$b^2$平凡），$K_G$的每个颜色置换自同构都是仿射映射。我们还证明（对$G$无任何限制），$K_G$的每个颜色置换自同构都是群自同构与保色图自同构的复合。这是2013年由D.P. Byrne、M.J. Donner和T.Q. Sibley猜想的。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
