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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
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    References listed on IDEAS

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    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.
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