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Neighbor sum distinguishing total colorings of IC-planar graphs with maximum degree 13

Author

Listed:
  • Chao Song

    (Hebei University of Technology)

  • Changqing Xu

    (Hebei University of Technology)

Abstract

A graph is IC-planar if it admits a drawing on the plane with at most one crossing per edge, such that two pairs of crossing edges share no common end vertex. For a given graph G, a proper total coloring $$\phi $$ϕ : $$V(G)~\cup ~E(G)\rightarrow \{1,2,\ldots ,k\}$$V(G)∪E(G)→{1,2,…,k} is called neighbor sum distinguishing if $$f_{\phi }(u)\ne f_{\phi }(v)$$fϕ(u)≠fϕ(v) for each $$uv\in E(G)$$uv∈E(G), where $$f_{\phi }(u)$$fϕ(u) is the sum of the color of u and the colors of the edges incident with u. The smallest integer k in such a coloring of G is the neighbor sum distinguishing total chromatic number, denoted by $$\chi ''_{\Sigma }(G)$$χΣ′′(G). Pilśniak and Woźniak conjectured $$\chi _{\Sigma }''(G)\le \Delta (G)+3$$χΣ′′(G)≤Δ(G)+3 for any simple graph with maximum degree $$\Delta (G)$$Δ(G). This conjecture was confirmed for IC-planar graph with maximum degree at least 14. In this paper, by using the discharging method, we prove that this conjecture holds for any IC-planar graph G with $$\Delta (G)=13$$Δ(G)=13.

Suggested Citation

  • Chao Song & Changqing Xu, 2020. "Neighbor sum distinguishing total colorings of IC-planar graphs with maximum degree 13," Journal of Combinatorial Optimization, Springer, vol. 39(1), pages 293-303, January.
  • Handle: RePEc:spr:jcomop:v:39:y:2020:i:1:d:10.1007_s10878-019-00467-1
    DOI: 10.1007/s10878-019-00467-1
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    References listed on IDEAS

    as
    1. Xin Zhang & Jianfeng Hou & Guizhen Liu, 2015. "On total colorings of 1-planar graphs," Journal of Combinatorial Optimization, Springer, vol. 30(1), pages 160-173, July.
    2. 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.
    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.
    4. 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.
    5. 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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    Cited by:

    1. Donghan Zhang, 2021. "Neighbor Sum Distinguishing Total Choosability of IC-Planar Graphs without Theta Graphs Θ 2,1,2," Mathematics, MDPI, vol. 9(7), pages 1-11, March.

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