TITLE: chromatic number of a graph versus its complement
QUESTION [9 upvotes]: What can be said about the rate of growth of $f(n)$, defined by 
$$f(n) = \min_{|V(G)|=n} \left[ \chi(G) + \chi(\bar{G}) \right],$$ where the minimum is taken over all graphs $G$ on $n$ vertices.
Two observations.
(1)  Either $G$ or $\bar{G}$ contains a clique on roughly $\log{n}$ vertices by Ramsey theory, so $f(n) \ge c_1 \log n $ for some constant $c_1 > 0$.
(2)  If $G = G(n,1/2)$ is a random graph, then $\chi(G) \approx \chi(\bar{G}) \approx n / \log{n}$ almost surely, so we also have $f(n) \le c_2 \,  n/ \log n$ for some constant $c_2 > 0$.
These bounds seem hopelessly far apart.

Can we improve on the bounds $$ c_1 \log n \le f(n) \le c_2 \, n / \log n$$ for all sufficiently large $n$?

REPLY [6 votes]: Although there is already an accepted answer, I answer to give a bit more information, for what I have to say would not fit in a comment.
This is the Nordhaus-Gaddum Theorem:

If $G$ is a graph of order $n$, then

$2 \sqrt{n} \leq \chi(G) + \chi(\bar{G}) \leq n+1$
$n \leq \chi(G) \cdot \chi(\bar{G}) \leq \left(\frac{n+1}{2}\right)^2$


And, there is no possible improvement of any of these bounds.  In fact, much more can be said:

Let $n$ be a positive integer.  For every two positive integers $a$
  and $b$ such that

$2 \sqrt{n} \leq a + b \leq n+1$
$n \leq a b \leq \left(\frac{n+1}{2}\right)^2$

there is a graph $G$ of order $n$ such that $\chi(G) = a$ and
  $\chi(\bar{G}) = b$.

Source: Graphs & Digraphs (Fifth edition) by Chartrand, Lesniak, and Zhang