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 $n$?