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The adjacent vertex distinguishing total choosability of planar graphs with maximum degree at least eleven

Author

Listed:
  • Xiaohan Cheng

    (University of Jinan)

  • Jianliang Wu

    (Shandong University)

Abstract

A (proper) total-k-coloring of a graph G is a mapping $$\phi : V (G) \cup E(G)\mapsto \{1, 2, \ldots , k\}$$ ϕ : V ( G ) ∪ E ( G ) ↦ { 1 , 2 , … , k } such that any two adjacent or incident elements in $$V (G) \cup E(G)$$ V ( G ) ∪ E ( G ) receive different colors. Let C(v) denote the set of the color of a vertex v and the colors of all incident edges of v. An adjacent vertex distinguishing total-k-coloring of G is a total-k-coloring of G such that for each edge $$uv\in E(G)$$ u v ∈ E ( G ) , $$C(u)\ne C(v)$$ C ( u ) ≠ C ( v ) . We denote the smallest value k in such a coloring of G by $$\chi ^{\prime \prime }_{a}(G)$$ χ a ″ ( G ) . It is known that $$\chi _{a}^{\prime \prime }(G)\le \Delta (G)+3$$ χ a ″ ( G ) ≤ Δ ( G ) + 3 for any planar graph with $$\Delta (G)\ge 10$$ Δ ( G ) ≥ 10 . In this paper, we consider the list version of this coloring and show that if G is a planar graph with $$\Delta (G)\ge 11$$ Δ ( G ) ≥ 11 , then $${ ch}_{a}^{\prime \prime }(G)\le \Delta (G)+3$$ c h a ″ ( G ) ≤ Δ ( G ) + 3 , where $${ ch}^{\prime \prime }_a(G)$$ c h a ″ ( G ) is the adjacent vertex distinguishing total choosability.

Suggested Citation

  • Xiaohan Cheng & Jianliang Wu, 2018. "The adjacent vertex distinguishing total choosability of planar graphs with maximum degree at least eleven," Journal of Combinatorial Optimization, Springer, vol. 35(1), pages 1-13, January.
  • Handle: RePEc:spr:jcomop:v:35:y:2018:i:1:d:10.1007_s10878-017-0149-6
    DOI: 10.1007/s10878-017-0149-6
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    References listed on IDEAS

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    1. Haiying Wang, 2007. "On the adjacent vertex-distinguishing total chromatic numbers of the graphs with Δ (G) = 3," Journal of Combinatorial Optimization, Springer, vol. 14(1), pages 87-109, July.
    2. 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.
    3. 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.
    4. 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.
    5. 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. Yulin Chang & Qiancheng Ouyang & Guanghui Wang, 2019. "Adjacent vertex distinguishing total choosability of planar graphs with maximum degree at least 10," Journal of Combinatorial Optimization, Springer, vol. 38(1), pages 185-196, July.
    2. Dan Yi & Junlei Zhu & Lixia Feng & Jiaxin Wang & Mengyini Yang, 2019. "Optimal r-dynamic coloring of sparse graphs," Journal of Combinatorial Optimization, Springer, vol. 38(2), pages 545-555, August.

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