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 \(tmc(G)-mc(G)=n-1\).
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Gao, Y., Li, P. & Li, X. Further results on the total monochromatic connectivity of graphs. J Comb Optim 44, 603–616 (2022). https://doi.org/10.1007/s10878-022-00850-5
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DOI: https://doi.org/10.1007/s10878-022-00850-5