Ringel-Youngs Theorem

The Ringel-Youngs theorem, known before its proof as the Heawood conjecture, gives the minimum number of colors needed to color every map on a surface of a given genus. For surfaces of genus 0, 1, 2, 3 and higher the required number of colors follows a formula giving 4, 7, 8, 9, 10 and so on; P. J. Heawood proposed the conjecture in 1890, and Gerhard Ringel and J. W. T. Youngs proved it in 1968, with the Klein bottle standing as the formula's sole exception, since Philip Franklin had shown in 1930 that it needs only six colors rather than the seven the general formula predicts. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/