# Characterization Results for theL(2, 1, 1)-Labeling Problem on Trees

• Volume: 37, Issue: 3, page 611-622
• ISSN: 2083-5892

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## Abstract

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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.

## How to cite

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Xiaoling Zhang, and Kecai Deng. "Characterization Results for theL(2, 1, 1)-Labeling Problem on Trees." Discussiones Mathematicae Graph Theory 37.3 (2017): 611-622. <http://eudml.org/doc/288292>.

@article{XiaolingZhang2017,
abstract = {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.},
author = {Xiaoling Zhang, Kecai Deng},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {L(2, 1, 1)-labeling; tree; diameter},
language = {eng},
number = {3},
pages = {611-622},
title = {Characterization Results for theL(2, 1, 1)-Labeling Problem on Trees},
url = {http://eudml.org/doc/288292},
volume = {37},
year = {2017},
}

TY - JOUR
AU - Xiaoling Zhang
AU - Kecai Deng
TI - Characterization Results for theL(2, 1, 1)-Labeling Problem on Trees
JO - Discussiones Mathematicae Graph Theory
PY - 2017
VL - 37
IS - 3
SP - 611
EP - 622
AB - 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.
LA - eng
KW - L(2, 1, 1)-labeling; tree; diameter
UR - http://eudml.org/doc/288292
ER -

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