A sufficient condition for a tree to be $$(\Delta +1)$$ ( Δ + 1 ) - $$(2,1)$$ ( 2, 1 ) -totally labelable

Zhengke Miao (), Qiaojun Shu, Weifan Wang and Dong Chen
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Zhengke Miao: Jiangsu Normal University
Qiaojun Shu: Zhejiang Normal University
Weifan Wang: Zhejiang Normal University
Dong Chen: Zhejiang Normal University

Journal of Combinatorial Optimization, 2016, vol. 31, issue 2, No 28, 893-901

Abstract: Abstract The $$(2, 1)$$ ( 2 , 1 ) -total labeling number $$\lambda _2^t(G)$$ λ 2 t ( G ) of a graph $$G$$ G is the width of the smallest range of integers that suffices to label the vertices and the edges of $$G$$ G such that no two adjacent vertices have the same label, no two adjacent edges have the same label and the difference between the labels of a vertex and its incident edges is at least $$2$$ 2 . It is known that every tree $$T$$ T with maximum degree $$\Delta $$ Δ has $$\Delta + 1 \le \lambda _2^t(T)\le \Delta + 2$$ Δ + 1 ≤ λ 2 t ( T ) ≤ Δ + 2 . In this paper, we give a sufficient condition for a tree $$T$$ T to have $$\lambda _2^t(T) = \Delta + 1$$ λ 2 t ( T ) = Δ + 1 . More precisely, we show that if $$T$$ T is a tree with $$\Delta \ge 4$$ Δ ≥ 4 and every $$\Delta $$ Δ -vertex in $$T$$ T is adjacent to at most $$\Delta - 3$$ Δ - 3 $$\Delta $$ Δ -vertices, then $$\lambda _2^t(T) = \Delta + 1$$ λ 2 t ( T ) = Δ + 1 . The result is best possible in the sense that there exist infinitely many trees $$T$$ T with $$\Delta \ge 4$$ Δ ≥ 4 and $$\lambda _2^t(T) = \Delta + 2$$ λ 2 t ( T ) = Δ + 2 such that each $$\Delta $$ Δ -vertex is adjacent to at most $$\Delta -2$$ Δ - 2 $$\Delta $$ Δ -vertices and at least one $$\Delta $$ Δ -vertex is adjacent to exactly $$\Delta -2$$ Δ - 2 vertices.

Keywords: $$(2; 1)$$ ( 2; 1 ) -Total labeling; Tree; Maximum degree (search for similar items in EconPapers)
Date: 2016
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DOI: 10.1007/s10878-014-9794-1

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