  A graph is said to be {\it total-colored} if all the edges and the vertices
of the graph are colored. A total-coloring of a graph is a {\it total
monochromatically-connecting coloring} ({\it TMC-coloring}, for short) if any
two vertices of the graph are connected by a path whose edges and internal
vertices on the path have the same color. For a connected graph $G$, the {\it
total monochromatic connection number}, denoted by $tmc(G)$, is defined as the
maximum number of colors used in a TMC-coloring of $G$. Note that a
TMC-coloring does not exist if $G$ is not connected, in which case we simply
let $tmc(G)=0$. In this paper, we first characterize all graphs of order $n$
and size $m$ with $tmc(G)=3,4,5,6,m+n-2,m+n-3$ and $m+n-4$, respectively. Then
we determine the threshold function for a random graph to have $tmc(G)\geq
f(n)$, where $f(n)$ is a function satisfying $1\leq f(n)<\frac{1}{2}n(n-1)+n$.
Finally, we show that for a given connected graph $G$, and a positive integer
$L$ with $L\leq m+n$, it is NP-complete to decide whether $tmc(G)\geq L$.
