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Total coloring of planar graphs without adjacent chordal 6-cycles

Author

Listed:
  • Huijuan Wang

    (Qingdao University)

  • Bin Liu

    (Ocean University of China)

  • Xiaoli Wang

    (Ocean University of China)

  • Guangmo Tong

    (University of Texas at Dallas)

  • Weili Wu

    (University of Texas at Dallas)

  • Hongwei Gao

    (Qingdao University)

Abstract

A total coloring of a graph G is a coloring such that no two adjacent or incident elements receive the same color. In this field there is a famous conjecture, named Total Coloring Conjecture, saying that the the total chromatic number of each graph G is at most $$\Delta +2$$ Δ + 2 . Let G be a planar graph with maximum degree $$\Delta \ge 7$$ Δ ≥ 7 and without adjacent chordal 6-cycles, that is, two cycles of length 6 with chord do not share common edges. In this paper, it is proved that the total chromatic number of G is $$\Delta +1$$ Δ + 1 , which partly confirmed Total Coloring Conjecture.

Suggested Citation

  • Huijuan Wang & Bin Liu & Xiaoli Wang & Guangmo Tong & Weili Wu & Hongwei Gao, 2017. "Total coloring of planar graphs without adjacent chordal 6-cycles," Journal of Combinatorial Optimization, Springer, vol. 34(1), pages 257-265, July.
    • Handle: RePEc:spr:jcomop:v:34:y:2017:i:1:d:10.1007_s10878-016-0063-3
    DOI: 10.1007/s10878-016-0063-3
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    References listed on IDEAS

    as
    1. Huijuan Wang & Lidong Wu & Weili Wu & Panos Pardalos & Jianliang Wu, 2014. "Minimum total coloring of planar graph," Journal of Global Optimization, Springer, vol. 60(4), pages 777-791, December.
    2. Hua Cai & Jianliang Wu & Lin Sun, 2016. "Total coloring of planar graphs without short cycles," Journal of Combinatorial Optimization, Springer, vol. 31(4), pages 1650-1664, May.
    3. Hualong Li & Laihao Ding & Bingqiang Liu & Guanghui Wang, 2015. "Neighbor sum distinguishing total colorings of planar graphs," Journal of Combinatorial Optimization, Springer, vol. 30(3), pages 675-688, October.
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    Cited by:

    1. Liting Wang & Huijuan Wang & Weili Wu, 2023. "Minimum total coloring of planar graphs with maximum degree 8," Journal of Combinatorial Optimization, Springer, vol. 45(2), pages 1-11, March.

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