Prime distance graphs and applications

Stimulated by the famous coloring task of R × R Eggleton coined the term distance graph and studied widely the prime distance graphs. A prime distance graph (PDG) G(Z, D) is one whose vertex set V is Z and the distance set D consists of a set of prime numbers. The edge set E(G) = {(u, v): u, ν ϵ V(G)}. It is characterized by the property that d(u, v) ϵ D where d(u, v) is the absolute difference of u and v. In this paper we prove certain existence and non-existence results concerning PDGs. We also discuss certain applications besides raising some open problems.