Chromatic number of graphs with special distance sets-III

Abstract

An integer distance graph is a graph G(Z;D) with the set of integers as vertex set and an edge joining two vertices u and v if and only if| u − v |∈ D where D is a subset of the positive integers. It is known that(G(Z; P)) = 4 where P is a set of Prime numbers. In this paper we have consideredthe open problem of characterizing class three sets when the distanceset D is not only a subset of primes P but also a special class of primes like Markov primes, Bell primes, Dihedral primes, Mills primes, Ramanujan primes,Quartan primes, Isolated primes and Thabit Number primes. We also indicatealternative formulations for a prime distance graph and raise certain open problems.