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Neighbor Sum Distinguishing Total Choosability of IC-Planar Graphs without Theta Graphs Θ 2,1,2

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  • Donghan Zhang

    (School of Mathematics and Statistics and Xi’an-Budapest Joint Research Center for Combinatorics, Northwestern Polytechnical University, Xi’an 710129, China
    College of Mathematics and Computer Applications, Shangluo University, Shangluo 726000, China)

Abstract

A theta graph Θ 2 , 1 , 2 is a graph obtained by joining two vertices by three internally disjoint paths of lengths 2, 1, and 2. A neighbor sum distinguishing (NSD) total coloring ϕ of G is a proper total coloring of G such that ∑ z ∈ E G ( u ) ∪ { u } ϕ ( z ) ≠ ∑ z ∈ E G ( v ) ∪ { v } ϕ ( z ) for each edge u v ∈ E ( G ) , where E G ( u ) denotes the set of edges incident with a vertex u . In 2015, Pilśniak and Woźniak introduced this coloring and conjectured that every graph with maximum degree Δ admits an NSD total ( Δ + 3 ) -coloring. In this paper, we show that the listing version of this conjecture holds for any IC-planar graph with maximum degree Δ ≥ 9 but without theta graphs Θ 2 , 1 , 2 by applying the Combinatorial Nullstellensatz, which improves the result of Song et al.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:7:p:708-:d:523783
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    References listed on IDEAS

    as
    1. 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.
    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. 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.
    4. 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.
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