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

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.Comment: 6 page

Chromatic Numbers Of Integer Distance Graphs

. An integer distance graph is a graph G(D) with the set of integers as vertex set and with an edge joining two vertices u and v if and only if ju \Gamma vj 2 D where D is a subset of the positive integers. We determine the chromatic number (D) of G(D) if D is a 4-element set of the form D = fx; y; x + y; y \Gamma xg; x ! y, or if D is an arithmetical progression D = fa + kd : k = 0; 1; 2; : : :g. 1. Introduction If S is a subset of the d-dimensional Euclidean space, S ` IR d , then the distance graph G(S; D) is defined as the graph G with vertex set V (G) = S and two vertices u and v are adjacent if and only if their distance d(u; v) is an element of the so-called distance set D which is a subset of the set of positive real numbers, D ` IR + . In case that V (G) = ZZ, the set of all integers, and D is a subset of the set IN of positive integers, D ` IN, the graph G(ZZ; D) = G(D) is called integer distance graph. A coloring f : V (G) ! ff 1 ; f 2 ; : : :g of G is an assignment..