Improved upper bounds for six-valent integer distance graph coloring periods

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, in which case the graphs have degree 6. Integer distance graphs can be defined equivalently as Cayley graphs on the group of integers under addition. In 1990 Eggleton, Erdős, and Skilton proved that if an integer distance graph of finite degree admits a proper k-coloring, then it admits a periodic proper k-coloring. They obtained an upper bound on the minimum such period but point out that it is quite large and very likely can be reduced considerably. In previous work, the authors of the present paper develop a general matrix method to approach the problem of finding chromatic numbers of abelian Cayley graphs. In this article we show that Zhu's theorem can be recovered as a special case of these results, and that in so doing we significantly improve the upper bounds on the periods of optimal colorings of these graphs.