A characterization of -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 + ∪ { 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 λ j , k ( G ) . Let T be any tree of maximum degree Δ and let d ≥ 2 be a positive integer. Then, for every c ∈ { 1 , … , min { Δ , d } } , T is in class c if λ d , 1 ( T ) = Δ + d + c − 2 . We characterize the class c of trees for every such c and also show that this class is non-empty.