Given a simple graph G (V, E) and a positive number d, an Ld(2, 1)-labelling of G is a function f V(G) [0, oc) such that whenever x, y E V are adjacent, If(x)-f(Y)l >-2d, and whenever the distance between x and y is two, If(x) f(Y)l >d. The Ld(2, 1)-labelling number A(G, d) is the smallest number m such that G has an Ld(2, 1)-labelling f with max{f(v) v E V} m. It is shown that to determine A(G, d), it suffices to study the case when d 1 and the labelling is nonnegative integral-valued. Let A(G) A(G, 1). The labelling numbers of special classes of graphs, e.g., A(C) 4 for any cycle C, are described. It is shown that for graphs of maximum degree A, A(G) _ A + 2A. If G is diameter 2, A(G) _ A2, a sharp bound for some A. Determining A(G) is shown to be NP-complete by relating it to the problem of finding Hamilton paths.

For positive integers, is the smallest k such that there exists a k-L(d 1 , d 2 )-labeling of G. This class of labelings is motivated by the code (or frequency) assignment problem in computer network. This article surveys the results on this labeling problem.

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For nonnegative integersis the smallest k such that there exists an L(d 1 , d 2 )-labeling with the largest label k. These labelings have an application to a computer code assignment problem. The task is to assign integer "control codes" to a network of computer stations with distance restrictions, which allow d 1 Յ d 2 . In this article, we will study the labelings with (d 1 , d 2 ) ʦ