On prime distance labeling of graphs

According to J.D. Laison, C. Starr and A. Walker a graph G is a prime distance garph if there exists a 1-1 labeling f : V(G) → Ζ such that for any two adjacent vertices u and ν the integer |f(u)-f(v)| is a prime. Further they called such a labelling of V(G) a prime distance labelling (PDL) of G. In this paper we prove certain existence and non-existence results concerning PDL. We also discuss certain applications besides raising some open problems.