On (p, 1)-total labelling of planar graphs

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• 作者：Lin Sun ; Jian-Liang Wu
• 关键词：(p ; 1) ; total labelling ; Minimal counterexample ; Discharging method
• 刊名：Journal of Combinatorial Optimization
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：33
• 期：1
• 页码：317-325
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Combinatorics; Convex and Discrete Geometry; Mathematical Modeling and Industrial Mathematics; Theory of Computation; Optimization; Operation Research/Decision Theory;
• 出版者：Springer US
• ISSN：1573-2886
• 卷排序：33

文摘

A k-(p, 1)-total labelling of a graph G is a function f from \(V(G)\cup E(G)\) to the color set \(\{0, 1, \ldots , k\}\) such that \(|f(u)-f(v)|\ge 1\) if \(uv\in E(G), |f(e_1)-f(e_2)|\ge 1\) if \(e_1\) and \(e_2\) are two adjacent edges in G and \(|f(u)-f(e)|\ge p\) if the vertex u is incident with the edge e. The minimum k such that G has a k-(p, 1)-total labelling, denoted by \(\lambda _p^T(G)\), is called the (p, 1)-total labelling number of G. In this paper, we prove that, for any planar graph G with maximum degree \(\Delta \ge 4p+4\) and \(p\ge 2, \lambda _p^T(G)\le \Delta +2p-2\).