$$$$-labeling number of Cartesian product of path and cycle |
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Authors: | Qiong?Wu Email author" target="_blank">Wai?Chee?ShiuEmail author Pak?Kiu?Sun |
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Institution: | 1.Department of Mathematics,Hong Kong Baptist University,Hong Kong,China;2.Department of Computational Science,Tianjin University of Technology and Education,Tianjin,China |
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Abstract: | For positive numbers \(j\) and \(k\), an \(L(j,k)\)-labeling \(f\) of \(G\) is an assignment of numbers to vertices of \(G\) such that \(|f(u)-f(v)|\ge j\) if \(d(u,v)=1\), and \(|f(u)-f(v)|\ge k\) if \(d(u,v)=2\). The span of \(f\) is the difference between the maximum and the minimum numbers assigned by \(f\). The \(L(j,k)\)-labeling number of \(G\), denoted by \(\lambda _{j,k}(G)\), is the minimum span over all \(L(j,k)\)-labelings of \(G\). In this article, we completely determine the \(L(j,k)\)-labeling number (\(2j\le k\)) of the Cartesian product of path and cycle. |
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