Characterization Results for the L(2, 1, 1)-Labeling Problem on Trees

An L(2, 1, 1)-labeling of a graph G is an assignment of non-negative integers (labels) to the vertices of G such that adjacent vertices receive labels with difference at least 2, and vertices at distance 2 or 3 receive distinct labels. The span of such a labelling is the difference between the maximum and minimum labels used, and the minimum span over all L(2, 1, 1)-labelings of G is called the L(2, 1, 1)-labeling number of G, denoted by λ_2,1,1(G). It was shown by King, Ras and Zhou in [The L(h, 1, 1)-labelling problem for trees, European J. Combin. 31 (2010) 1295–1306] that every tree T has Δ₂(T) − 1 ≤ λ_2,1,1(T) ≤ Δ₂(T), where Δ₂(T) = max_uv∈E(T)(d(u) + d(v)). And they conjectured that almost all trees have the L(2, 1, 1)-labeling number attain the lower bound. This paper provides some sufficient conditions for λ_2,1,1(T) = Δ₂(T). Furthermore, we show that the sufficient conditions we provide are also necessary for trees with diameter at most 6.