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

  • Junlei Zhu
  • Yuehua Bu
  • Miltiades P. Pardalos
  • Hongwei Du
  • Huijuan Wang
  • Bin Liu
Article
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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(uv) 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.

Keywords

Planar graph Cycle Labeling 

Notes

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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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Junlei Zhu
    • 1
    • 2
  • Yuehua Bu
    • 1
    • 3
  • Miltiades P. Pardalos
    • 4
  • Hongwei Du
    • 5
  • Huijuan Wang
    • 6
  • Bin Liu
    • 7
  1. 1.Department of MathematicsZhejiang Normal UniversityJinhuaChina
  2. 2.College of Mathematics, Physics and Information EngineeringJiaxing UniversityJiaxingChina
  3. 3.Zhejiang Normal University Xingzhi CollegeJinhuaChina
  4. 4.Department of Industrial and System EngineeringUniversity of FloridaGainesvilleUSA
  5. 5.Department of Computer Science and TechnologyHarbin Institute of Technology Shenzhen Graduate SchoolShenzhenChina
  6. 6.School of Mathematics and StatisticsQingdao UniversityQingdaoChina
  7. 7.Department of MathematicsOcean University of ChinaQingdaoChina

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