Given a map with genus 
, Heawood showed in 1890 that the maximum number 
of colors necessary to color a map
(the chromatic number) on an unbounded surface
is
 |  |  |
(1)
|
 |  |  |
(2)
|
where 
is the floor function, 
is the genus, and 
is the Euler characteristic.
This is the Heawood conjecture. In 1968, for
any unbounded orientable surface other than the sphere
(or equivalently, the plane) and any nonorientable surface
other than the Klein bottle, 
was shown to be not merely a maximum, but the actual number
needed (Ringel and Youngs 1968).
When the four-color theorem was proven, the Heawood formula was shown to hold also for all orientable and nonorientable unbounded
surfaces with the exception of the Klein bottle.
For the Klein bottle only, the actual number of colors
needed is six--one less than 
(Franklin 1934; Saaty 1986, p. 45).
The Möbius strip, which is a bounded surface,
also requires 6 colors, while blind application of the Heawood formula (which is
not applicable in this case) gives 7.
| surface |  |  |  |
| Klein bottle | 0 | 7 | 6 |
| Möbius strip | 0 | 7 | 6 |
| plane | 2 | 4 | 4 |
| projective plane | 1 | 6 | 6 |
| sphere | 2 | 4 | 4 |
| torus | 0 | 7 | 7 |
