IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v11y2023i10p2233-d1143699.html
   My bibliography  Save this article

Adjacent Vertex Distinguishing Coloring of Fuzzy Graphs

Author

Listed:
  • Zengtai Gong

    (College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China
    These authors contributed equally to this work.)

  • Chen Zhang

    (College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China
    School of Mathematics and Information Engineering, Longdong University, Qingyang 745000, China
    These authors contributed equally to this work.)

Abstract

In this paper, we consider the adjacent vertex distinguishing proper edge coloring (for short, AVDPEC) and the adjacent vertex distinguishing total coloring (for short, AVDTC) of a fuzzy graph. Firstly, this paper describes the development process, the application areas, and the existing review research of fuzzy graphs and adjacent vertex distinguishing coloring of crisp graphs. Secondly, we briefly introduce the coloring theory of crisp graphs and the related theoretical basis of fuzzy graphs, and add some new classes of fuzzy graphs. Then, based on the α -cuts of fuzzy graphs and distance functions, we give two definitions of the AVDPEC of fuzzy graphs, respectively. A lower bound on the chromatic number of the AVDPEC of a fuzzy graph is obtained. With examples, we show that some results of the AVDPEC of a crisp graph do not carry over to our set up; the adjacent vertex distinguishing chromatic number of the fuzzy graph is different from the general chromatic number of a fuzzy graph. We also give a simple algorithm to construct a ( d , f ) -extended AVDPEC for fuzzy graphs. After that, in a similar way, two definitions of the AVDTC of fuzzy graphs are discussed. Finally, the future research directions of distinguishing coloring of fuzzy graphs are given.

Suggested Citation

  • Zengtai Gong & Chen Zhang, 2023. "Adjacent Vertex Distinguishing Coloring of Fuzzy Graphs," Mathematics, MDPI, vol. 11(10), pages 1-25, May.
  • Handle: RePEc:gam:jmathe:v:11:y:2023:i:10:p:2233-:d:1143699
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/11/10/2233/pdf
    Download Restriction: no

    File URL: https://www.mdpi.com/2227-7390/11/10/2233/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Hervé Hocquard & Mickaël Montassier, 2013. "Adjacent vertex-distinguishing edge coloring of graphs with maximum degree Δ," Journal of Combinatorial Optimization, Springer, vol. 26(1), pages 152-160, July.
    2. 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)

    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. 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.
    2. 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.
    3. Zhuoya Liu & Changqing Xu, 2022. "Adjacent vertex distinguishing edge coloring of IC-planar graphs," Journal of Combinatorial Optimization, Springer, vol. 43(4), pages 710-726, May.
    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. 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.
    6. 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.
    7. 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.
    8. Yi Wang & Jian Cheng & Rong Luo & Gregory Mulley, 2016. "Adjacent vertex-distinguishing edge coloring of 2-degenerate graphs," Journal of Combinatorial Optimization, Springer, vol. 31(2), pages 874-880, February.
    9. Junlei Zhu & Yuehua Bu & Yun Dai, 2018. "Upper bounds for adjacent vertex-distinguishing edge coloring," Journal of Combinatorial Optimization, Springer, vol. 35(2), pages 454-462, February.
    10. 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.
    11. 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.
    12. 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.
    13. Joanna Skowronek-Kaziów, 2017. "Graphs with multiplicative vertex-coloring 2-edge-weightings," Journal of Combinatorial Optimization, Springer, vol. 33(1), pages 333-338, January.

    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:gam:jmathe:v:11:y:2023:i:10:p:2233-:d:1143699. 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: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.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.