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Optimal channel assignment and L(p, 1)-labeling

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Abstract

The optimal channel assignment is an important optimization problem with applications in optical networks. This problem was formulated to the L(p, 1)-labeling of graphs by Griggs and Yeh (SIAM J Discrete Math 5:586–595, 1992). A k-L(p, 1)-labeling of a graph G is a function \(f:V(G)\rightarrow \{0,1,2,\ldots ,k\}\) such that \(|f(u)-f(v)|\ge p\) if \(d(u,v)=1\) and \(|f(u)-f(v)|\ge 1\) if \(d(u,v)=2\), where d(u, v) is the distance between the two vertices u and v in the graph. Denote \(\lambda _{p,1}^l(G)= \min \{k \mid G\) has a list k-L(p, 1)-labeling\(\}\). In this paper we show upper bounds \(\lambda _{1,1}^l(G)\le \Delta +9\) and \(\lambda _{2,1}^l(G)\le \max \{\Delta +15,29\}\) for planar graphs G without 4- and 6-cycles, where \(\Delta \) is the maximum vertex degree of G. Our proofs are constructive, which can be turned to a labeling (channel assignment) method to reach the upper bounds.

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Acknowledgements

This work was supported in part by National Natural Science Foundation of China (11501316, 11771403) and Shandong Provincial Natural Science Foundation of China (ZR2017QA010, ZR2017MF055).

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Authors and Affiliations

  1. Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, Zhejiang, China

    Junlei Zhu & Yuehua Bu

  2. College of Mathematics, Physics and Information Engineering, Jiaxing University, Jiaxing, 314001, China

    Junlei Zhu

  3. Zhejiang Normal University Xingzhi College, Jinhua, 321004, Zhejiang, China

    Yuehua Bu

  4. Department of Industrial and System Engineering, University of Florida, Gainesville, FL, 32611, USA

    Miltiades P. Pardalos

  5. Department of Computer Science and Technology, Harbin Institute of Technology Shenzhen Graduate School, Shenzhen, China

    Hongwei Du

  6. School of Mathematics and Statistics, Qingdao University, Qingdao, 266100, China

    Huijuan Wang

  7. Department of Mathematics, Ocean University of China, Qingdao, 266100, China

    Bin Liu

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  1. Junlei Zhu
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  2. Yuehua Bu
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  3. Miltiades P. Pardalos
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  4. Hongwei Du
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  5. Huijuan Wang
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  6. Bin Liu
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Correspondence to Huijuan Wang.

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Zhu, J., Bu, Y., Pardalos, M.P. et al. Optimal channel assignment and L(p, 1)-labeling. J Glob Optim 72, 539–552 (2018). https://doi.org/10.1007/s10898-018-0647-9

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  • DOI: https://doi.org/10.1007/s10898-018-0647-9

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