Chromatic number of prime distance graphs (Q1329821)

For any set \(D\) of positive integers, the distance graph \(G(D)\) is the graph with the vertex set \(Z\) and edge set \(\{(u, v) : | u-v | \in D \}\). The authors consider the problem of determining all minimal sets \(D\) of primes such that \(G(D)\) is 4-chromatic. This problem, originally proposed by \textit{R. B. Eggleton, P. Erdős} and \textit{D. K. Skilton} [Graphs Comb. 6, No. 1, 17-32 (1990; Zbl 0698.05033)], is solved in the present paper for 4-element sets \(D\).