size of maximal independent set and chromatic number

Let $\alpha (G)$ be the size of the largest independent set in a graph $G$, and $\chi (G)$ the chromatic number of $G$.

Theorem: $\alpha (G)\chi (G)\ge |G|$.

Proof.

The vertices of $G$ can be partitioned into $\chi (G)$ monochromatic classes. Each class is an independent set, and hence cannot have size larger than $\alpha (G)$.∎