Abstract
The \((2, 1)\)-total labeling number \(\lambda _2^t(G)\) of a graph \(G\) is the width of the smallest range of integers that suffices to label the vertices and the edges of \(G\) 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 \(2\). It is known that every tree \(T\) with maximum degree \(\Delta \) has \(\Delta + 1 \le \lambda _2^t(T)\le \Delta + 2\). In this paper, we give a sufficient condition for a tree \(T\) to have \(\lambda _2^t(T) = \Delta + 1\). More precisely, we show that if \(T\) is a tree with \(\Delta \ge 4\) and every \(\Delta \)-vertex in \(T\) is adjacent to at most \(\Delta - 3\) \(\Delta \)-vertices, then \(\lambda _2^t(T) = \Delta + 1\). The result is best possible in the sense that there exist infinitely many trees \(T\) with \(\Delta \ge 4\) and \(\lambda _2^t(T) = \Delta + 2\) such that each \(\Delta \)-vertex is adjacent to at most \(\Delta -2\) \(\Delta \)-vertices and at least one \(\Delta \)-vertex is adjacent to exactly \(\Delta -2\) vertices.
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Acknowledgments
Zhengke Miao’s Research supported by NSFC (No. 11171288). Weifan Wang’s Research supported by NSFC (No. 11371328)
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Miao, Z., Shu, Q., Wang, W. et al. A sufficient condition for a tree to be \((\Delta +1)\)-\((2,1)\)-totally labelable. J Comb Optim 31, 893–901 (2016). https://doi.org/10.1007/s10878-014-9794-1
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DOI: https://doi.org/10.1007/s10878-014-9794-1