Chromatic number of prime distance graphs

AbstractFor any set D of positive integers, the distance graph G(D)G(V,E) is the graph with vertex set V(G)Z and edge set E(G){(u,v):|u−v|∈D}. In Research Problem 77 (Discrete Math. 69 (1988) 105–106) Eggleton, Erdös and Skilton propose the problem to determine all minimal subsets D of the prime numbers such that graph G(D) is 4-chromatic. In the present paper this problem is solved for 4-element prime sets D