Self-complementary graph implies chromatic number is at least equal to square root of size of vertex set

Statement

Suppose $G$ is a finite undirected graph on $n$ vertices. Suppose further that $G$ is a self-complementary graph. Then, the chromatic number of $G$ is at least ${\sqrt {n}}$.

Facts used

1. Clique number is less than or equal to chromatic number
2. Independence number times chromatic number is greater than or equal to size of vertex set
3. Independence number equals clique number of complement