Abstract
A k-(2, 1)-total labelling of a graph G is a mapping \(f: V(G)\cup E(G)\rightarrow \{0,1,\ldots ,k\}\) such that adjacent vertices or adjacent edges receive distinct labels, and a vertex and its incident edges receive labels that differ in absolute value by at least 2. The (2, 1)-total number, denoted \(\lambda _2^t(G)\), is the minimum k such that G has a k-(2, 1)-total labelling. Let T be a tree with maximum degree \(\Delta \ge 7\). A vertex \(v\in V(T)\) is called major if \(d(v)=\Delta \), minor if \(d(v)<\Delta \), and saturated if v is major and is adjacent to exactly \(\Delta - 2\) major vertices. It is known that \(\Delta + 1 \le \lambda _2^t(T)\le \Delta + 2\). In this paper, we prove that if every major vertex is adjacent to at most \(\Delta -2\) major vertices, and every minor vertex is adjacent to at most three saturated vertices, then \(\lambda _2^t(T) = \Delta + 1\). The result is best possible with respect to these required conditions.
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Q. Shu: Research supported partially by ZJNSFC LQ15A010010.
W. Wang: Research supported partially by NSFC(No.11071223).
Y. Wang: Research supported partially by NSFC(No.11301035).
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Shu, Q., Wang, W. & Wang, Y. A new sufficient condition for a tree T to have the (2, 1)-total number \(\Delta +1\) . J Comb Optim 33, 1011–1020 (2017). https://doi.org/10.1007/s10878-016-0021-0
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DOI: https://doi.org/10.1007/s10878-016-0021-0