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

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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 \(\Delta \ge 7\).

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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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Authors and Affiliations

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

    Huijuan Wang

  2. Department of Computer Science, University of Texas at Tyler, Tyler, TX, 75799, USA

    Lidong Wu

  3. Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL, 32611-6595, USA

    Miltiades P. Pardalos

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

    Hongwei Du

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

    Bin Liu

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  1. Huijuan Wang
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  2. Lidong Wu
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  3. Miltiades P. Pardalos
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  4. Hongwei Du
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  5. Bin Liu
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Correspondence to Bin Liu.

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Wang, H., Wu, L., Pardalos, M.P. et al. An efficient case for computing minimum linear arboricity with small maximum degree. Optim Lett 13, 419–428 (2019). https://doi.org/10.1007/s11590-018-1278-2

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  • DOI: https://doi.org/10.1007/s11590-018-1278-2

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