L(2, 1)-labellings of integer distance graphs Peter Che Bor Lam, 1 Tao-Ming Wang 1, 2 and Guohua Gu 3 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) ≤ Δ 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. 1 Introduction The channel assignment problem (FAP) is is the assignment of frequencies to television and radio transmitters subject to restrictions imposed by the distance between transmit- ters. This problem was first formulated as a graph coloring problem by Hale [4]. In 1988, Roberts (in a private communication to Griggs) introduced a variation of this problem, where “close” transmitters must receive different channels and “very close” transmitters must receive channels at least two apart. Motivated by this variation, Griggs and Yeh [6] first proposed and studied the L(2, 1)-labelling of a simple graph with a condition at distance two. This is followed by many other works. For examples, see [1–3,5,8,10,12,13]. An L(2, 1)-labelling f of G is a function f : V (G) → [0,k], such that |f (u) - f (v)|≥ 2 if uv ∈ E(G); and |f (u) -f (v)|≥ 1 if d G (u, v) = 2, where d G (u, v) is the length (number of edges) of a shortest path between u and v in G. Elements of the image under f are called labels, and the span of f , denoted by span(f ), is the difference between the maximum and minimum labels of f . Without loss of generality, we assume that the minimum label of 1 BNU-HKBU International College, Zhuhai, P. R. China; part of the work was done while visiting Department of Mathematics, Tunghai University, Taichung, Taiwan 2 Partially supported by National Science Council grant NSC 95-2115-M-029-003 3 Department of Mathematics, Southeast University, Nanjing 210018, P. R. China 1