The L(2,1)-labelling number of distance graphs G(D), denoted by λ(D), isstudied. It is shown that distance graphs satisfy λ(G) ≤Δ~2. Moreover, we prove λ({1,2, ..., k})=2k +2 and λ({1,3,..., 2k -1}) =2k + 2 for any fixed positive integer k. Suppose k, a ∈ N and k,a≥2. If k≥a, then λ({a, a + 1,..., a + k - 1}) = 2(a + k-1). Otherwise, λ({a, a + 1, ..., a + k- 1}) ≤min{2(a + k-1), 6k -2}. When D consists of two positive integers,6≤λ(D)≤8. For thespecial distance sets D = {k, k + 1}(any k ∈N), the upper bound of λ(D) is improved to 7.
An integer distance graph is a graph G(Z, D) with the integer set Z as vertexset, in which an edge joining two vertices u and v if and only if | u - v | ∈ D, where D is a setof natural numbers. Using a related theorem in combinatorics and some conclusions known to us in thecoloring of the distance graph, the chromatic number _X(G) is determined in this paper that is ofthe distance graph G(Z, D) for some finite distance sets D containing {2, 3} with D = 4 andcontaining {2, 3, 5} with | D | = 5 by the method in which the combination of a few periodiccolorings.
