Suppose that on a surface with Euler characteristic
amap with
facescan be coloured by at most
colourswhere 
Write this inequality as
If
thensome faces can be coloured twice only if they have no edges incommon. We consider the least value offorvalues of
If
then
sothat at least 7 colours are needed.
If %XHI =1 then
Proceeding in this way, we obtain the table
|
|
|
| 2 | 6 |
| 1 | 6 |
| 0 | 7 |
| -1 | 7 |
| -2 | 8 |
| -3 | 9 |
| -4 | 9 |
| -5 | 10 |
| -6 | 10 |
| -7 | 10 |
| -8 | 11 |
| -9 | 11 |
| -10 | 12 |
