L(2, 1)-labellings of integer distance graphs

Peter Che Bor Lam, Tao-Ming Wang and Guohua Gu Abstract Let D be a set of positive integers. The (integer) distance graph G(Z,D) with distance set D is the graph with vertex set Z, in which two vertices x, y are adjacent if and only if |x − y| ∈ D. An L(2, 1)-labelling of a graph G is an assignment of nonnegative integers to the vertices of G such that labels of any two adjacent vertices differ by at least 2, and labels of any two vertices that are at distance two apart are distinct. The minimum range of labels over all L(2, 1)-labellings of a graph G is called the L(2, 1)−labelling number, or simply the λ-number of G, and is denoted by λ(G). We use λ(D) to denote the λ-number of G(Z,D). In this paper, some bounds for λ(D) are established. It is also shown that distance graphs satisfy the conjecture λ(G) ≤ ∆2. We also use a periodic labelling and prove that there exists an algorithm to determine the labelling number for any distance graph with finite distance set. For some special distance sets D, better upper bounds for λ(D) are obtained. We shall also determine the exact values of λ(D) for some two element set D.