Distance Graphs Generated by Five Primes (Research)

Let D be a set of prime numbers. We investigate the chromatic number of graphs with vertex set of the integers and edges between any pair of vertices whose absolute difference falls in D. Such a graph is called a prime distance graph, and the set D is called the distance set. The chromatic number of prime distance graphs is known when the distance set D has at most four primes. In this paper we begin to classify prime distance graphs with a distance set of five primes. In particular, we completely classify the family of distance sets D = {2, 3, 7, 19, p} where p is any prime, and solve most of the more general family D = {2, 3, 7, p, q} for any primes p and q. The number theoretic function κ(D) is used as a tool, and some general properties about κ(D) are established.