I'm working on the following graph theory excercise:
Let $G$ be a simple graph with $n$ vertices, which is regular of degree $d$. By considering the number of vertices that can be assigned the same colour, prove that $\chi(G) ≥ n/(n - d)$, being $\chi(G)$ the chromatic number of $G$
I'm starting from the theorem that says: For every graph $G$:
$$\chi(G) ≤ 1 + \Delta(G)$$
As $\Delta(G) = d$, then
$$\chi(G) ≤ 1 + d$$
and $n/n = 1$ then
$$\chi(G) ≤ n/n + d$$
What am I missing to fit my procedure into the way requested? Thanks in advance for any hint or help.
Suppose $\chi (G) < \frac {n}{n-d} \iff n-d<\frac { n}{\chi (G) }$ . Then we have one color $k$ such that at least $n-d+1$ vertices have that color. Then all these vertices are non adjacent. So degree of any one such vertex $\leq d-1$ which is a contradiction.