Chromatic number of graphs with special distance sets, I

Abstract

Given a subset \(D\) of positive integers, an integer distance graph is a graph \(G(\mathbb{Z}, D)\) with the set \(\mathbb{Z}\) of integers as vertex set and with an edge joining two vertices \(u\) and \(v\) if and only if \(|u - v| \in D\). In this paper we consider the problem of determining the chromatic number of certain integer distance graphs \(G(\mathbb{Z}, D)\)whose distance set \(D\) is either 1) a set of \((n+1)\) positive integers for which the \(n^{th}\) power of the last is the sum of the \(n^{th}\) powers of the previous terms, or 2) a set of pythagorean quadruples, or 3) a set of pythagorean \(n\)-tuples, or 4) a set of square distances, or 5) a set of abundant numbers or deficient numbers or carmichael numbers, or 6) a set of polytopic numbers, or 7) a set of happy numbers or lucky numbers, or 8) a set of Lucas numbers, or 9) a set of \(\mathcal{U}\)lam numbers, or 10) a set of weird numbers. Besides finding the chromatic number of a few specific distance graphs we also give useful upper and lower bounds for general cases. Further, we raise some open problems.