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 λj,k-number for G, denoted by λj,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 [3], we present upper and lower bounds for λj,k(T ) in terms of j, k, M(T ), and ∆(T ) (the maximum degree of T ). For a special case when j > ∆(T )k, the upper and the lower bounds are k apart. Moreover, we completely determine λj,k(T ) for trees T with j > M(T )k.