AI 中文总结
本文引入单项式Cayley图研究单位方向代数关系对平面色数的影响，确定单位根情形的色数，并证明几何嵌入图三可着色性，提出代数参数三色猜想。
AI 中文摘要
单位方向之间的代数关系如何影响平面的色数？我们引入单项式Cayley图作为研究这一问题的框架，首先考虑允许的单位步长为单个复参数的带符号幂的图。超越参数给出二部图，而对于单位根，我们确定了其色数和循环色数。作为几何推论，每个允许在平面中进行单位距离嵌入且任意两条边方向之间的夹角为$\pi$的有理倍数的图都是三可着色的。对于代数非整数参数，四种颜色足够，而当本原整数最小多项式的首项系数至少为3时，三种颜色足够。我们猜想对于每个代数非整数方向参数，三种颜色足够，而对于每个方向参数，四种颜色足够。
英文摘要
How do algebraic relations among unit directions influence the chromatic number of the plane? We introduce monomial Cayley graphs as a framework for investigating this question, beginning with graphs whose allowed unit steps are signed powers of a single complex parameter. Transcendental parameters give bipartite graphs, while for roots of unity we determine both the chromatic and circular chromatic numbers. As a geometric consequence, every graph admitting a unit-distance embedding in the plane in which the angle between any two edge directions is a rational multiple of $π$ is three-colorable. For algebraic noninteger parameters, four colors suffice, and three suffice when the primitive integer minimal polynomial has leading coefficient at least three. We conjecture that three colors suffice for every algebraic noninteger direction parameter and that four suffice for every direction parameter.
Comments50 pages; ancillary data and verification scripts included
