L(j, k)-labelling and maximum ordering-degrees for trees

Let G be a graph. For two vertices u and v in G, we denote d(u, v) the distance between u and v. Let j, k be positive integers with j >= k. An L(j, k)-labelling for G is a function f : V(G) -> {0, 1, 2, ...} such that for any two vertices u and v, vertical bar f(u) - f(v)vertical bar is at least j if d(u, v) = 1; and is at least k if d(u, v) = 2. The span of f is the difference between the largest and the smallest numbers in f (V). The lambda(j,k)-number for G, denoted by lambda(j,k)(G), is the minimum span over all L(j, k)-labellings of G. We introduce a new parameter for a tree T, namely, the maximum ordering-degree, denoted by M(T). Combining this new parameter and the special family of infinite trees introduced by Chang and Lu (2003) [3], we present upper and lower bounds for lambda(j,k)(T) in terms of j, k, M(T), and Delta(T) (the maximum degree of T). For a special case when j >= Delta(T)k, the upper and the lower bounds are k apart. Moreover, we completely determine lambda(j,k)(T) for trees T with j >= M(T)k. (C) 2009 Elsevier B.V. All rights reserved.