A class of full (d, 1)-colorable trees

Let j and k be nonnegative integers. An L(j, k)-labeling of a graph G, where j ≥ k, is 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 d(u, v) = 2, 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 f of G, denoted λj,k(G), is called the span of G. An L(j, k)-labeling of G that has a span of λj,k(G) is called a span labeling of G. We say that G is (j, k)-full colorable, denoted (j, k)-FC, if there exists a span labeling f of G such that the set {i | f−1({i}) = ∅, where 1 ≤ i ≤ span(f)−1} = ∅. Fishburn and Roberts showed (in SIAM J. Discrete Math. 20 (2006), 428–443) that if T is a tree of order n ≥ Δ(T ) + 2, then T is (2, 1)-FC. In this paper, we show that there exists a class of (d, 1)-FC trees where d ≥ 3.