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Optimization Letters

, Volume 13, Issue 2, pp 419–428 | Cite as

An efficient case for computing minimum linear arboricity with small maximum degree

  • Huijuan Wang
  • Lidong Wu
  • Miltiades P. Pardalos
  • Hongwei Du
  • Bin LiuEmail author
Original Paper
  • 33 Downloads

Abstract

Graph coloring has interesting applications in optimization, calculation of Hessian matrix, network design and so on. In this paper, we consider an improper edge coloring which is one important coloring-linear arboricity. For a graph G, a linear forest is a disjoint union of paths and cycles. The linear arboricity la(G) is the minimum number of disjoint linear forests such that their union is exactly the edge set of G. In this paper, we study a special case that G is a simple planar graph with two not adjacent cycles each with a chordal and length between 4 and 7. We show that in this special case, \(la(G)=\lceil \frac{\Delta }{2}\rceil \) where \(\Delta \) is the maximum vertex degree of G and \(\Delta \ge 7\).

Keywords

Linear arboricity Cycles Planar graph 

Notes

Acknowledgements

This work was supported in part by National Natural Science Foundation of China (Grant Nos. 11501316, 71171120, 71571180), China Postdoctoral Science Foundation (2016M600556), Qingdao Postdoctoral Application Research Project (2016156), Shandong Provincial Natural Science Foundation of China (Grant Nos. ZR2017QA010, ZR2017MF055).

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Huijuan Wang
    • 1
  • Lidong Wu
    • 2
  • Miltiades P. Pardalos
    • 3
  • Hongwei Du
    • 4
  • Bin Liu
    • 5
    Email author
  1. 1.School of Mathematics and StatisticsQingdao UniversityQingdaoChina
  2. 2.Department of Computer ScienceUniversity of Texas at TylerTylerUSA
  3. 3.Department of Industrial and Systems EngineeringUniversity of FloridaGainesvilleUSA
  4. 4.Department of Computer Science and Technology, Harbin Institute of TechnologyShenzhen Graduate SchoolShenzhenChina
  5. 5.Department of MathematicsOcean University of ChinaQingdaoChina

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