IDEAS home Printed from https://ideas.repec.org/a/spr/jcomop/v39y2020i2d10.1007_s10878-019-00480-4.html
   My bibliography  Save this article

On the total neighbour sum distinguishing index of graphs with bounded maximum average degree

Author

Listed:
  • H. Hocquard

    (University of Bordeaux)

  • J. Przybyło

    (AGH University of Science and Technology)

Abstract

A proper total k-colouring of a graph $$G=(V,E)$$G=(V,E) is an assignment $$c : V \cup E\rightarrow \{1,2,\ldots ,k\}$$c:V∪E→{1,2,…,k} of colours to the edges and the vertices of G such that no two adjacent edges or vertices and no edge and its end-vertices are associated with the same colour. A total neighbour sum distinguishing k-colouring, or tnsd k-colouring for short, is a proper total k-colouring such that $$\sum _{e\ni u}c(e)+c(u)\ne \sum _{e\ni v}c(e)+c(v)$$∑e∋uc(e)+c(u)≠∑e∋vc(e)+c(v) for every edge uv of G. We denote by $$\chi ''_{\Sigma }(G)$$χΣ′′(G) the total neighbour sum distinguishing index of G, which is the least integer k such that a tnsd k-colouring of G exists. It has been conjectured that $$\chi ''_{\Sigma }(G) \le \Delta (G) + 3$$χΣ′′(G)≤Δ(G)+3 for every graph G. In this paper we confirm this conjecture for any graph G with $$\mathrm{mad}(G)

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jcomop:v:39:y:2020:i:2:d:10.1007_s10878-019-00480-4
    DOI: 10.1007/s10878-019-00480-4
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10878-019-00480-4
    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/s10878-019-00480-4?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. 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.
    2. 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.
    Full references (including those not matched with items on IDEAS)

    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. 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.
    2. 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.
    3. 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.
    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. 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.
    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. 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.
    8. 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.
    9. 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.
    10. 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.
    11. 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.
    12. 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.
    13. 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.

    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:39:y:2020:i:2:d:10.1007_s10878-019-00480-4. 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.