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International Conference on Algorithmic Applications in Management

AAIM 2020: Algorithmic Aspects in Information and Management pp 469–480Cite as

Total Coloring of Outer-1-planar Graphs: The Cold Case

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12290))

Abstract

A graph is outer-1-planar if it has a drawing in the plane so that its vertices are on the boundary face and each edge is crossed at most once. Zhang (2013) proved that the total chromatic number of every outer-1-planar graph with maximum degree \(\varDelta \ge 5\) is \(\varDelta +1\), and showed that there are graphs with maximum degree 3 and total chromatic number 5. For outer-1-planar graphs with maximum degree 4, Zhang (2017) confirmed that its total chromatic number is at most 5 if it admits an outer-1-planar drawing in the plane so that any two pairs of crossing edges share at most one common end vertex. In this paper, we prove that the total chromatic number of every Anicop graph with maximum degree 4 is at most 5, where an Anicop graph is an outer-1-planar graph that admits a drawing in the plane so that if there are two pairs of crossing edges sharing two common end vertices, then any of those two pairs of crossing edges would not share any end vertex with some other pair of crossing edges. This result generalizes the one of Zhang (2017) and moves a step towards the complete solving of the cold case.

Supported by NSFC (11871055) and the Youth Talent Support Plan of Xi’an Association for Science and Technology (2018-6).

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Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Xidian University, Xi’an, 710071, China

    Weichan Liu & Xin Zhang

  2. Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing, China

    Weichan Liu

  3. University of Chinese Academy of Sciences, Beijing, China

    Weichan Liu

Authors
  1. Weichan Liu
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  2. Xin Zhang
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Corresponding author

Correspondence to Xin Zhang .

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Editors and Affiliations

  1. Zhejiang Normal University, Jinhua, China

    Zhao Zhang

  2. Texas Southern University, Houston, TX, USA

    Wei Li

  3. University of Texas at Dallas, Richardson, TX, USA

    Ding-Zhu Du

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Liu, W., Zhang, X. (2020). Total Coloring of Outer-1-planar Graphs: The Cold Case. In: Zhang, Z., Li, W., Du, DZ. (eds) Algorithmic Aspects in Information and Management. AAIM 2020. Lecture Notes in Computer Science(), vol 12290. Springer, Cham. https://doi.org/10.1007/978-3-030-57602-8_42

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  • DOI: https://doi.org/10.1007/978-3-030-57602-8_42

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-57601-1

  • Online ISBN: 978-3-030-57602-8

  • eBook Packages: Computer ScienceComputer Science (R0)

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