AI 中文总结
研究具有不可缺少生成元的循环图色数问题，利用Heuberger矩阵理论，给出三个生成元（其中一个不可缺少）循环图色数上界，给出Garcia - Marco和Knauer定理阿贝尔群情形的另证，更系统证明Heuberger关于两个生成元循环图色数的定理。
AI 中文摘要
凯莱图$\text{Cay}(G,S)$的顶点集是群$G$，当且仅当$xy^{-1}$或$yx^{-1}$属于$G$的某个固定子集$S$时，两个顶点$x$和$y$相邻，$S$的元素称为生成元。循环图是$G$为有限循环群的凯莱图。许多作者研究过循环图的色数。当$S$有两个元素时，已知Heuberger给出的循环图色数的一般公式，但当$S$有三个或更多元素时未知。若$S\setminus\{x\}$不能生成$G$，则称$S$中的元素$x$是不可缺少的；若$S$的每个元素都是不可缺少的，则称$S$是最小的。根据Garcia - Marco和Knauer 2024年的结果，若$G$是幂零群且$S$是最小的，则$\text{Cay}(G,S)$是3 - 可着色的。本文证明了三个主要结果：一是给出了具有三个生成元（其中一个不可缺少）的循环图色数的上界；二是给出了Garcia - Marco和Knauer定理在阿贝尔群情形下的另一种证明；三是应用这些方法为Heuberger关于具有两个生成元的循环图色数的定理提供了更系统（且可能可推广）的证明。本文主要工具是Heuberger矩阵理论，并提供了简要介绍。
英文摘要
The Cayley graph $\text{Cay}(G,S)$ is the graph whose vertex set is the group $G$, where two vertices $x$ and $y$ are adjacent if and only if $xy^{-1}$ or $yx^{-1}$ lies in some fixed subset $S$ of $G$. We call the elements of $S$ generators. A circulant graph is a Cayley graph where $G$ is finite and cyclic. Chromatic numbers of circulant graphs have been studied by many authors. A general formula due to Heuberger for the chromatic number of a circulant graph is known when $S$ has two elements, but no such formula is known when $S$ has three or more elements. We say that an element $x$ of $S$ is indispensable if $S\setminus\{x\}$ does not generate $G$. We say that $S$ is minimal if every element of $S$ is indispensable. By a result of Garcia-Marco and Knauer from 2024, if $G$ is nilpotent and $S$ is minimal, then $\text{Cay}(G,S)$ is $3$-colorable. In this article, we prove three main results. First, we give an upper bound for the chromatic number of a circulant graph with three generators, one of which is indispensable. Second, we present an alternate proof of the theorem of Garcia-Marco and Knauer for the case of abelian groups. Third, we apply these methods to provide a considerably more systematic (and potentially generalizable) proof of Heuberger's theorem for the chromatic number of circulant graphs with two generators. Throughout this paper, our primary tool is the theory of Heuberger matrices, for which we provide a brief primer.
