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 λ2,1,1(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 ≤ λ2,1,1(T) ≤ Δ2(T), where Δ2(T) = maxuv∈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 λ2,1,1(T) = Δ2(T). Furthermore, we show that the sufficient conditions we provide are also necessary for trees with diameter at most 6.
@article{bwmeta1.element.doi-10_7151_dmgt_1935,
author = {Xiaoling Zhang and Kecai Deng},
title = {Characterization Results for theL(2, 1, 1)-Labeling Problem on Trees},
journal = {Discussiones Mathematicae Graph Theory},
volume = {37},
year = {2017},
pages = {611-622},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_7151_dmgt_1935}
}
Xiaoling Zhang; Kecai Deng. Characterization Results for theL(2, 1, 1)-Labeling Problem on Trees. Discussiones Mathematicae Graph Theory, Tome 37 (2017) pp. 611-622. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_7151_dmgt_1935/