A characterization of lambdad, 1-minimal trees and other attainable classes

An L(j,k)-labeling of a graph G, where j>=k, is defined as a function f:V(G)->Z^[email protected]?{0} such that if u and v are adjacent vertices in G, then |f(u)-f(v)|>=j, while if u and v are vertices such that the length of the shortest path joining them is two, then |f(u)-f(v)|>=k. The largest label used by f is the span of f. The smallest span among all L(j,k)-labelings of G is denoted by @l"j","k(G). Let T be any tree of maximum degree @D and let d>=2 be a positive integer. Then, for every [email protected]?{1,...,min{@D,d}}, T is in class c if @l"d","1(T)[email protected]+d+c-2. We characterize the class c of trees for every such c and also show that this class is non-empty.