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Total coloring of planar graphs without adjacent short cycles

Author

Listed:
  • Huijuan Wang

    (Qingdao University)

  • Bin Liu

    (Ocean University of China)

  • Yan Gu

    (Qingdao University)

  • Xin Zhang

    (Xidian University)

  • Weili Wu

    (TaiYuan University of Technology
    University of Texas at Dallas)

  • Hongwei Gao

    (Qingdao University)

Abstract

In the study of computer science, optimization, computation of Hessians matrix, graph coloring is an important tool. In this paper, we consider a classical coloring, total coloring. Let $$G=(V,E)$$ G = ( V , E ) be a graph. Total coloring is a coloring of $$V\cup {E}$$ V ∪ E such that no two adjacent or incident elements (vertex/edge) receive the same color. Let G be a planar graph with $$\varDelta \ge 8$$ Δ ≥ 8 . We proved that if for every vertex $$v\in V$$ v ∈ V , there exists two integers $$i_v,j_v\in \{3,4,5,6,7\}$$ i v , j v ∈ { 3 , 4 , 5 , 6 , 7 } such that v is not incident with adjacent $$i_v$$ i v -cycles and $$j_v$$ j v -cycles, then the total chromatic number of graph G is $$\varDelta +1$$ Δ + 1 .

Suggested Citation

  • Huijuan Wang & Bin Liu & Yan Gu & Xin Zhang & Weili Wu & Hongwei Gao, 2017. "Total coloring of planar graphs without adjacent short cycles," Journal of Combinatorial Optimization, Springer, vol. 33(1), pages 265-274, January.
    • Handle: RePEc:spr:jcomop:v:33:y:2017:i:1:d:10.1007_s10878-015-9954-y
    DOI: 10.1007/s10878-015-9954-y
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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. 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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