TITLE: G is a graph with n vertices. Prove that $χ(G) · χ( \overline G) ≥ n$.
QUESTION [5 upvotes]: Let $G$ be a graph with n vertices. Prove that $\chi(G)\cdot\chi(\overline G)\geqslant n$.

I know that the chromatic number of a graph with $n$ vertices will have somewhere between $\chi(G-\ni) \leqslant\chi(G)\leqslant\ni(G-v)+1$ and also that the upperbound of the chromatic number will be greater than or equal tot he max degree of a vertex in $G+1$ and that the lower bound is the greater than or equal to the largest clique.
Is there anything we can do with this information to prove our proposition?

REPLY [5 votes]: Let $n$ be the number of vertices of $G$. Take then an $a$-coloring of $G$ and a $b$-coloring of $\overline{G}$. Try to produce from that a coloring of the union of $G$ and $\overline{G}$ (and note that it is a complete graph on $n$ vertices).