IDEAS home Printed from https://ideas.repec.org/a/spr/jcomop/v30y2015i3d10.1007_s10878-013-9660-6.html
   My bibliography  Save this article

Neighbor sum distinguishing total colorings of planar graphs

Author

Listed:
  • Hualong Li

    (Shandong University)

  • Laihao Ding

    (Shandong University)

  • Bingqiang Liu

    (Shandong University)

  • Guanghui Wang

    (Shandong University)

Abstract

A total [k]-coloring of a graph $$G$$ G is a mapping $$\phi : V (G) \cup E(G)\rightarrow [k]=\{1, 2,\ldots , k\}$$ ϕ : V ( G ) ∪ E ( G ) → [ k ] = { 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 $$f(v)$$ f ( v ) denote the sum of the color of a vertex $$v$$ v and the colors of all incident edges of $$v$$ v . A total $$[k]$$ [ k ] -neighbor sum distinguishing-coloring of $$G$$ G is a total $$[k]$$ [ k ] -coloring of $$G$$ G such that for each edge $$uv\in E(G)$$ u v ∈ E ( G ) , $$f(u)\ne f(v)$$ f ( u ) ≠ f ( v ) . By $$\chi ^{''}_{nsd}(G)$$ χ n s d ′ ′ ( G ) , we denote the smallest value $$k$$ k in such a coloring of $$G$$ G . Pilśniak and Woźniak conjectured $$\chi _{nsd}^{''}(G)\le \Delta (G)+3$$ χ n s d ′ ′ ( G ) ≤ Δ ( G ) + 3 for any simple graph with maximum degree $$\Delta (G)$$ Δ ( G ) . In this paper, we prove that this conjecture holds for any planar graph with maximum degree at least 13.

Suggested Citation

  • 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.
  • Handle: RePEc:spr:jcomop:v:30:y:2015:i:3:d:10.1007_s10878-013-9660-6
    DOI: 10.1007/s10878-013-9660-6
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10878-013-9660-6
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.

    File URL: https://libkey.io/10.1007/s10878-013-9660-6?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    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. 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.
    3. 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.
    4. You Lu & Chuandong Xu & Zhengke Miao, 2018. "Neighbor sum distinguishing list total coloring of subcubic graphs," Journal of Combinatorial Optimization, Springer, vol. 35(3), pages 778-793, April.
    5. Miaomiao Han & You Lu & Rong Luo & Zhengke Miao, 2018. "Neighbor sum distinguishing total coloring of graphs with bounded treewidth," Journal of Combinatorial Optimization, Springer, vol. 36(1), pages 23-34, July.
    6. H. Hocquard & J. Przybyło, 2020. "On the total neighbour sum distinguishing index of graphs with bounded maximum average degree," Journal of Combinatorial Optimization, Springer, vol. 39(2), pages 412-424, February.
    7. 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.
    8. Huijuan Wang & Bin Liu & Xiaoli Wang & Guangmo Tong & Weili Wu & Hongwei Gao, 2017. "Total coloring of planar graphs without adjacent chordal 6-cycles," Journal of Combinatorial Optimization, Springer, vol. 34(1), pages 257-265, July.
    9. Hongjie Song & Changqing Xu, 2017. "Neighbor sum distinguishing total coloring of planar graphs without 4-cycles," Journal of Combinatorial Optimization, Springer, vol. 34(4), pages 1147-1158, November.
    10. 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.
    11. 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.
    12. 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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. 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.
    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. 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.
    4. 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.
    5. 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.
    6. Zengtai Gong & Chen Zhang, 2023. "Adjacent Vertex Distinguishing Coloring of Fuzzy Graphs," Mathematics, MDPI, vol. 11(10), pages 1-25, May.
    7. 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.
    8. 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.
    9. 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.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:jcomop:v:30:y:2015:i:3:d:10.1007_s10878-013-9660-6. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.