Let $\Sigma_g$ be a surface of genus $g$. The Heawood Conjecture gives a closed formula in one variable, $\chi$ (the Euler characterstic of $\Sigma_g$), for the minimal number of of colors needed to color any graph drawn on $\Sigma_g$.
Ringel-Youngs, 1968: The Heawood Conjecture is true, except for $g=0$ (where we don't know the answer - this would be the Four-Color Conjecture) and the Klein bottle (where a graph was shown to be colorable with fewer than predicted by the formula).