User blog comment:Vel!/Call for math facts/@comment-5529393-20140730085241

Proving that the Heawood formula is an upper bound for the chromatic number of any closed surface except the sphere is provable in about half a page, and was proven by Heawood in 1890. The case of the sphere was not proven until 1975, and required a lengthy computer proof.

The Heawood formula is accurate for all closed surfaces except Klein bottle.