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Total coloring of planar graphs without short cycles

Author

Listed:
  • Hua Cai

    (Shandong University
    Changji University)

  • Jianliang Wu

    (Shandong University)

  • Lin Sun

    (Shandong University
    Changji University)

Abstract

The total chromatic number of a graph $$G$$ G , denoted by $$\chi ''(G)$$ χ ′ ′ ( G ) , is the minimum number of colors needed to color the vertices and edges of $$G$$ G such that no two adjacent or incident elements get the same color. It is known that if a planar graph $$G$$ G has maximum degree $$\Delta (G)\ge 9$$ Δ ( G ) ≥ 9 , then $$\chi ''(G)=\Delta (G)+1$$ χ ′ ′ ( G ) = Δ ( G ) + 1 . In this paper, it is proved that if $$G$$ G is a planar graph with $$\Delta (G)\ge 7$$ Δ ( G ) ≥ 7 , and for each vertex $$v$$ v , there is an integer $$k_v\in \{3,4,5,6,7,8\}$$ k v ∈ { 3 , 4 , 5 , 6 , 7 , 8 } such that there is no $$k_v$$ k v -cycle which contains $$v$$ v , then $$\chi ''(G)=\Delta (G)+1$$ χ ′ ′ ( G ) = Δ ( G ) + 1 .

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jcomop:v:31:y:2016:i:4:d:10.1007_s10878-015-9859-9
    DOI: 10.1007/s10878-015-9859-9
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    References listed on IDEAS

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    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.
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    Cited by:

    1. 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.

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