Zhu关于六值整数距离图定理的一个推广
A generalization of Zhu's theorem on six-valent integer distance graphs
AI总结：
本文证明Zhu关于三元整数距离图色数的定理是作者先前关于阿贝尔群上Cayley图色数一般方法（针对3×2矩阵情形）的特例，从而推广了该定理。
摘要：
给定一个正整数集合$S$，整数距离图以整数集为顶点集，其中两个顶点相邻当且仅当它们差的绝对值属于$S$。2002年，Zhu完全确定了当$S$的基数为$3$时整数距离图的色数。整数距离图可以等价地定义为整数加法群上的Cayley图。在之前的一篇论文中，作者们发展了通用方法来处理阿贝尔群上Cayley图色数的问题。对每个这样的图，关联一个整数矩阵。在某些情况下，色数可以直接由矩阵元素确定。特别地，作者们完全确定了当矩阵大小为$3\ imes 2$时的色数——这正是Zhu所研究的图所关联矩阵的大小。因此，在本文中，我们证明Zhu的定理可以作为作者们先前结果的一个特例被重新得到。
英文摘要：
Given a set $S$ of positive integers, the integer distance graph for $S$ has the set of integers as its vertex set, where two vertices are adjacent if and only if the absolute value of their difference lies in $S$. In 2002, Zhu completely determined the chromatic number of integer distance graphs when $S$ has cardinality $3$. Integer distance graphs can be defined equivalently as Cayley graphs on the group of integers under addition. In a previous paper, the authors develop general methods to approach the problem of finding chromatic numbers of Cayley graphs on abelian groups. To each such graph one associates an integer matrix. In some cases the chromatic number can be determined directly from the matrix entries. In particular, the authors completely determine the chromatic number whenever the matrix is of size $3\times 2$ -- precisely the size of the matrices associated to the graphs studied by Zhu. In this paper, then, we demonstrate that Zhu's theorem can be recovered as a special case of the authors' previous results.
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