IDEAS home Printed from https://ideas.repec.org/a/spr/jcomop/v44y2022i1d10.1007_s10878-022-00850-5.html
   My bibliography  Save this article

Further results on the total monochromatic connectivity of graphs

Author

Listed:
  • Yanhong Gao

    (Nankai University)

  • Ping Li

    (Nankai University)

  • Xueliang Li

    (Nankai University)

Abstract

The concepts of monochromatic connection number mc(G) (MC-number for short) and vertex monochromatic connection number mvc(G) (MVC-number for short) of a graph G were introduced in 2011 and 2018, respectively, by Caro and Yuster and Cai et al., and have been studied extensively, While in 2017, Jiang et al. introduced the concept of total monochromatic connection number tmc(G) (TMC-number for shot) of a graph G. In this paper, we mainly study the TMC-number of a graph. At first, we completely determine the TMC-numbers for any given simple and connected graphs, and obtain some Nordhaus-Gaddum-type results for the TMC-number. Jiang et al. in 2017 put forward a conjecture and a problem on the difference between tmc(G), mc(G) and mvc(G) of a graph G. We then completely solve the conjecture and the problem, and characterize the graphs G of order n with $$tmc(G)-mc(G)=n-1$$ t m c ( G ) - m c ( G ) = n - 1 .

Suggested Citation

  • Yanhong Gao & Ping Li & Xueliang Li, 2022. "Further results on the total monochromatic connectivity of graphs," Journal of Combinatorial Optimization, Springer, vol. 44(1), pages 603-616, August.
    • Handle: RePEc:spr:jcomop:v:44:y:2022:i:1:d:10.1007_s10878-022-00850-5
    DOI: 10.1007/s10878-022-00850-5
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10878-022-00850-5
    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/s10878-022-00850-5?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. Xueliang Li & Di Wu, 2017. "The (vertex-)monochromatic index of a graph," Journal of Combinatorial Optimization, Springer, vol. 33(4), pages 1443-1453, May.
    2. Jiao Zhou & Zhao Zhang & Weili Wu & Kai Xing, 2014. "A greedy algorithm for the fault-tolerant connected dominating set in a general graph," Journal of Combinatorial Optimization, Springer, vol. 28(1), pages 310-319, July.
    3. Yishuo Shi & Yaping Zhang & Zhao Zhang & Weili Wu, 2016. "A greedy algorithm for the minimum $$2$$ 2 -connected $$m$$ m -fold dominating set problem," Journal of Combinatorial Optimization, Springer, vol. 31(1), pages 136-151, January.
    4. Qingqiong Cai & Xueliang Li & Di Wu, 2018. "Some extremal results on the colorful monochromatic vertex-connectivity of a graph," Journal of Combinatorial Optimization, Springer, vol. 35(4), pages 1300-1311, May.
    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. Xiaozhi Wang & Xianyue Li & Bo Hou & Wen Liu & Lidong Wu & Suogang Gao, 2021. "A greedy algorithm for the fault-tolerant outer-connected dominating set problem," Journal of Combinatorial Optimization, Springer, vol. 41(1), pages 118-127, January.
    2. Zhao Zhang & Jiao Zhou & Shaojie Tang & Xiaohui Huang & Ding-Zhu Du, 2018. "Computing Minimum k -Connected m -Fold Dominating Set in General Graphs," INFORMS Journal on Computing, INFORMS, vol. 30(2), pages 217-224, May.
    3. Yaoyao Zhang & Zhao Zhang & Ding-Zhu Du, 2023. "Construction of minimum edge-fault tolerant connected dominating set in a general graph," Journal of Combinatorial Optimization, Springer, vol. 45(2), pages 1-12, March.
    4. Jiao Zhou & Zhao Zhang & Shaojie Tang & Xiaohui Huang & Ding-Zhu Du, 2018. "Breaking the O (ln n ) Barrier: An Enhanced Approximation Algorithm for Fault-Tolerant Minimum Weight Connected Dominating Set," INFORMS Journal on Computing, INFORMS, vol. 30(2), pages 225-235, May.
    5. Ping Li & Xueliang Li, 2022. "Monochromatic disconnection: Erdős-Gallai-type problems and product graphs," Journal of Combinatorial Optimization, Springer, vol. 44(1), pages 136-153, August.
    6. Kejia Zhang & Qilong Han & Guisheng Yin & Haiwei Pan, 2016. "OFDP: a distributed algorithm for finding disjoint paths with minimum total length in wireless sensor networks," Journal of Combinatorial Optimization, Springer, vol. 31(4), pages 1623-1641, May.
    7. Austin Buchanan & Je Sang Sung & Sergiy Butenko & Eduardo L. Pasiliao, 2015. "An Integer Programming Approach for Fault-Tolerant Connected Dominating Sets," INFORMS Journal on Computing, INFORMS, vol. 27(1), pages 178-188, February.
    8. Mao Luo & Huigang Qin & Xinyun Wu & Caiquan Xiong, 2024. "A novel local search approach with connected dominating degree-based incremental neighborhood evaluation for the minimum 2-connected dominating set problem," Journal of Combinatorial Optimization, Springer, vol. 47(5), pages 1-26, July.
    9. Yaoyao Zhang & Chaojie Zhu & Shaojie Tang & Yingli Ran & Ding-Zhu Du & Zhao Zhang, 2024. "Evolutionary Algorithm on General Cover with Theoretically Guaranteed Approximation Ratio," INFORMS Journal on Computing, INFORMS, vol. 36(2), pages 510-525, March.
    10. Huijuan Wang & Panos M. Pardalos & Bin Liu, 2019. "Optimal channel assignment with list-edge coloring," Journal of Combinatorial Optimization, Springer, vol. 38(1), pages 197-207, 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:44:y:2022:i:1:d:10.1007_s10878-022-00850-5. 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.