A generalization of Zhu's theorem on six-valent integer distance graphs

TL;DR

This paper generalizes Zhu's theorem on the chromatic number of six-valent integer distance graphs by framing it within a broader algebraic context involving Cayley graphs and matrix methods.

Contribution

The authors extend Zhu's theorem by developing a general approach using integer matrices to determine chromatic numbers of Cayley graphs on abelian groups.

Findings

Zhu's theorem is recovered as a special case of the new framework.

A matrix-based method for calculating chromatic numbers is established.

The approach applies to a broader class of graphs beyond Zhu's original setting.

Abstract

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.