More on total monochromatic connection of graphs

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 $tmc(G)\geq L$.