Heawood number

The Heawood number of a surface is an upper bound for the maximalnumber of colors needed to color any graph embedded in thesurface. In 1890 Heawood proved for all surfaces except the spherethat no more than

$$H(S)=\lfloor \frac{7+\sqrt{49-24e(S)}}{2}\rfloor$$

colors are needed to color any graph embedded in a surface ofEuler characteristic $e(S)$. The case of the sphere is the four-color conjecture which was settled by Appel and Haken in 1976. The number $H(S)$ becameknown as Heawood number in 1976. Franklin proved that the chromatic numberof a graph embedded in the Klein bottle can be as large as $6$,but never exceeds $6$. Later it was proved in the works of Ringeland Youngs that the complete graph of $H(S)$ vertices can beembedded in the surface $S$ unless $S$ is the Klein bottle. Thisestablished that Heawood’s bound could not be improved.

For example, the complete graph on $7$ vertices can be embedded in the torus as follows: