AI 中文总结
本文证明阿贝尔群连通凯莱图保持自然边着色的同构可分解为群同构与保色图自同构的复合，并给出保色置换自同构均为仿射映射的充要条件。
AI 中文摘要
我们证明，如果 $\varphi$ 是两个阿贝尔群连通凯莱图之间的同构，且 $\varphi$ 保持凯莱图的自然边着色，则 $\varphi$ 是一个群同构与一个保色图自同构的复合。这意味着，如果连通阿贝尔凯莱图 $Cay(G;S)$ 的每个保色自同构都是仿射映射，那么每个保色置换自同构也是如此。我们还证明，该性质成立当且仅当对 $G$ 中每个二阶元素 $c$，由 $\{\\, s \in S \mid 2s \neq c \\,\} \cup \{c\}$ 生成的子群的指数 $\le 2$。
英文摘要
We prove that if $φ$ is an isomorphism between two connected Cayley graphs of abelian groups, and $φ$ respects the natural edge-colourings of the Cayley graphs, then $φ$ is the composition of a group isomorphism and a colour-preserving graph automorphism. This implies that if every colour-preserving automorphism of a connected abelian Cayley graph $Cay(G;S)$ is an affine map, then the same is true for every colour-permuting automorphism. We also show that this property holds if and only if the subgroup generated by $\{\, s \in S \mid 2s \neq c \,\} \cup \{c\}$ has index $\le 2$ for every element $c$ of order $2$ in $G$.
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