$\text{option (C). }~m(n,r) = \binom{r}{2}$
An assumption that there’s some order of these $r$ colors,..
to newly selected vertex we’ll assign first color that has not already been assigned to any of its neighbors,
we’ll get a new color only if new node is connected to all the other vertices which have a distinct color assigned to them,
recursively to get $r^{th}$ color , new vertex should be connected to at-least all vertices having $(r-1)$ distinct colors,
minimum value of $m(n,r)$ is given by,
$T(r)=T(r-1) + (r-1) \\ T(1) = 0$
$T(r)=T(r-i) + (r – i) + (r-(i-1)) + … + (r-1) \\ \quad r-i = 1 \Rightarrow i = r -1$
$\text{substituting the value of i, we get}$
$T(r)=T(r-(r-1)) + (r – (r-1)) + (r-(r-2)) + … + (r-1) \\ \quad \quad \Rightarrow T(1) + 1 + 2+ … + (r-1) \\ \quad \quad \Rightarrow 0 + \frac{(r-1)(r-1+1)}{2} \\ \quad \quad \Rightarrow \frac{r(r-1)}{2} \\ \quad \quad \Rightarrow \binom{r}{c}$