Chromatic numbers of circulants with indispensable generators

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.