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 \(tmc(G)-mc(G)=n-1\).

图G的单色连接数mc ( G )（简称 MC-number）和顶点单色连接数mvc ( G )（简称 MVC-number）的概念分别由 Caro 和 Yuster 于 2011 年和 2018 年引入和蔡等人，并进行了广泛的研究，而在2017年，江等人。引入了图G的总单色连接数tmc ( G ) (TMC-number for shot) 的概念. 在本文中，我们主要研究图的 TMC 数。首先，我们完全确定任何给定的简单连通图的 TMC 数，并获得 TMC 数的一些 Nordhaus-Gaddum 类型结果。江等人。2017年对图G的tmc ( G )、mc ( G )和mvc ( G )的区别提出了一个猜想和一个问题。然后我们完全解决了猜想和问题，并用\(tmc(G)-mc(G)=n-1\)表征了n阶图 G。