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.
is given by the Heawood
conjecture,
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.
-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
