The number of colors sufficient for map coloring on a surface of genus g is given by the Heawood conjecture, χ(g) = ⌊1/2(7 + sqrt(48g + 1))⌋, where ⌊x⌋ is the floor function. The fact that χ(g) (which is called the chromatic number) is also necessary was proved by Ringel and Youngs with two exceptions: the sphere (which requires the same number of colors as the plane) and the Klein bottle.
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