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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.
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 -
