where
is the floor function. The fact that (which is called the chromatic
number) is also necessary was proved by Ringel
and Youngs (1968) with two exceptions: the sphere (which
requires the same number of colors as the plane) and the
Klein bottle.
A -holed torus
therefore requires
colors. For ,
1, ..., the first few values of are 4, 7 (illustrated above, M. Malak, pers. comm.,
Feb. 22, 2006), 8, 9, 10, 11, 12, 12, 13, 13, 14, 15, 15, 16, ... (OEIS A000934).
A set of regions requiring the maximum of seven regions is shown above for a normal
torus
The above figure shows the relationship between the Heawood
graph and the 7-color torus coloring.