IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v10y2022i6p912-d770093.html
   My bibliography  Save this article

Total Coloring of Dumbbell Maximal Planar Graphs

Author

Listed:
  • Yangyang Zhou

    (School of Electronics Engineering and Computer Science, Peking University, Beijing 100871, China
    Key Laboratory of High Confidence Software Technologies, Peking University, Beijing 100871, China)

  • Dongyang Zhao

    (School of Electronics Engineering and Computer Science, Peking University, Beijing 100871, China
    Key Laboratory of High Confidence Software Technologies, Peking University, Beijing 100871, China)

  • Mingyuan Ma

    (School of Electronics Engineering and Computer Science, Peking University, Beijing 100871, China
    Key Laboratory of High Confidence Software Technologies, Peking University, Beijing 100871, China)

  • Jin Xu

    (School of Electronics Engineering and Computer Science, Peking University, Beijing 100871, China
    Key Laboratory of High Confidence Software Technologies, Peking University, Beijing 100871, China)

Abstract

The Total Coloring Conjecture (TCC) states that every simple graph G is totally ( Δ + 2 ) -colorable, where Δ denotes the maximum degree of G . In this paper, we prove that TCC holds for dumbbell maximal planar graphs. Especially, we divide the dumbbell maximal planar graphs into three categories according to the maximum degree: J 9 , I-dumbbell maximal planar graphs and II-dumbbell maximal planar graphs. We give the necessary and sufficient condition for I-dumbbell maximal planar graphs, and prove that any I-dumbbell maximal planar graph is totally 8-colorable. Moreover, a linear time algorithm is proposed to compute a total ( Δ + 2 ) -coloring for any I-dumbbell maximal planar graph.

Suggested Citation

  • Yangyang Zhou & Dongyang Zhao & Mingyuan Ma & Jin Xu, 2022. "Total Coloring of Dumbbell Maximal Planar Graphs," Mathematics, MDPI, vol. 10(6), pages 1-10, March.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:6:p:912-:d:770093
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/10/6/912/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/10/6/912/
    Download Restriction: no
    ---><---

    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:10:y:2022:i:6:p:912-:d:770093. 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.

    We have no bibliographic references for this item. You can help adding them by using 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.