Integral Distance Graphs

Abstract

Suppose D is a subset of all positive integers. The distance graph G(Z, D) with distance set D is the graph with vertex set Z, and two vertices x and y are adjacent if and only if |x - y| ∈ D. This paper studies the chromatic number χ(Z, D) of G(Z, D). In particular, we prove that χ(Z, D) ≤ |D| + 1 when |D| is finite. Exact values of χ(G, D) are also determined for some D With |D| = 3. © 1997 John Wiley & Sons, Inc.