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, |f(u)-f(v)| 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 @l"j","k-number for G, denoted by @l"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 @l"j","k(T) in terms of j, k, M(T), and @D(T) (the maximum degree of T). For a special case when j>[email protected](T)k, the upper and the lower bounds are k apart. Moreover, we completely determine @l"j","k(T) for trees T with j>=M(T)k.