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The adjacent vertex distinguishing total coloring of planar graphs

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  • Weifan Wang

    (Zhejiang Normal University)

  • Danjun Huang

    (Zhejiang Normal University)

Abstract

An adjacent vertex distinguishing total coloring of a graph G is a proper total coloring of G such that any pair of adjacent vertices have distinct sets of colors. The minimum number of colors needed for an adjacent vertex distinguishing total coloring of G is denoted by $\chi''_{a}(G)$ . In this paper, we characterize completely the adjacent vertex distinguishing total chromatic number of planar graphs G with large maximum degree Δ by showing that if Δ≥14, then $\varDelta+1\leq \chi''_{a}(G)\leq \varDelta+2$ , and $\chi''_{a}(G)=\varDelta+2$ if and only if G contains two adjacent vertices of maximum degree.

Suggested Citation

  • Weifan Wang & Danjun Huang, 2014. "The adjacent vertex distinguishing total coloring of planar graphs," Journal of Combinatorial Optimization, Springer, vol. 27(2), pages 379-396, February.
    • Handle: RePEc:spr:jcomop:v:27:y:2014:i:2:d:10.1007_s10878-012-9527-2
    DOI: 10.1007/s10878-012-9527-2
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    References listed on IDEAS

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    1. Yiqiao Wang & Weifan Wang, 2010. "Adjacent vertex distinguishing total colorings of outerplanar graphs," Journal of Combinatorial Optimization, Springer, vol. 19(2), pages 123-133, February.
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    Cited by:

    1. Cunquan Qu & Guanghui Wang & Guiying Yan & Xiaowei Yu, 2016. "Neighbor sum distinguishing total choosability of planar graphs," Journal of Combinatorial Optimization, Springer, vol. 32(3), pages 906-916, October.
    2. Jingjing Yao & Xiaowei Yu & Guanghui Wang & Changqing Xu, 2017. "Neighbor sum distinguishing total coloring of 2-degenerate graphs," Journal of Combinatorial Optimization, Springer, vol. 34(1), pages 64-70, July.
    3. Renyu Xu & Jianliang Wu & Jin Xu, 2016. "Neighbor sum distinguishing total coloring of graphs embedded in surfaces of nonnegative Euler characteristic," Journal of Combinatorial Optimization, Springer, vol. 31(4), pages 1430-1442, May.
    4. Patcharapan Jumnongnit & Kittikorn Nakprasit, 2017. "Graphs with Bounded Maximum Average Degree and Their Neighbor Sum Distinguishing Total-Choice Numbers," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2017, pages 1-4, November.
    5. Tong Li & Cunquan Qu & Guanghui Wang & Xiaowei Yu, 2017. "Neighbor product distinguishing total colorings," Journal of Combinatorial Optimization, Springer, vol. 33(1), pages 237-253, January.
    6. Jingjing Huo & Yiqiao Wang & Weifan Wang & Wenjing Xia, 2020. "Neighbor-distinguishing total coloring of planar graphs with maximum degree twelve," Journal of Combinatorial Optimization, Springer, vol. 39(1), pages 246-272, January.
    7. Weifan Wang & Jingjing Huo & Danjun Huang & Yiqiao Wang, 2019. "Planar graphs with $$\Delta =9$$Δ=9 are neighbor-distinguishing totally 12-colorable," Journal of Combinatorial Optimization, Springer, vol. 37(3), pages 1071-1089, April.
    8. 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.
    9. Yang, Donglei & Sun, Lin & Yu, Xiaowei & Wu, Jianliang & Zhou, Shan, 2017. "Neighbor sum distinguishing total chromatic number of planar graphs with maximum degree 10," Applied Mathematics and Computation, Elsevier, vol. 314(C), pages 456-468.
    10. Lin Sun & Xiaohan Cheng & Jianliang Wu, 2017. "The adjacent vertex distinguishing total coloring of planar graphs without adjacent 4-cycles," Journal of Combinatorial Optimization, Springer, vol. 33(2), pages 779-790, February.
    11. Enqiang Zhu & Chanjuan Liu & Jiguo Yu, 2020. "Neighbor product distinguishing total colorings of 2-degenerate graphs," Journal of Combinatorial Optimization, Springer, vol. 39(1), pages 72-76, January.
    12. Xiaohan Cheng & Guanghui Wang & Jianliang Wu, 2017. "The adjacent vertex distinguishing total chromatic numbers of planar graphs with $$\Delta =10$$ Δ = 10," Journal of Combinatorial Optimization, Springer, vol. 34(2), pages 383-397, August.
    13. Xu, Changqing & Li, Jianguo & Ge, Shan, 2018. "Neighbor sum distinguishing total chromatic number of planar graphs," Applied Mathematics and Computation, Elsevier, vol. 332(C), pages 189-196.

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