Torus coloring is map coloring on a torus . More generally, for a connected closed surface
that is an orientable surface of genus chromatic
number over maps with connected regions, where adjacent regions share a boundary
arc, is
where floor function . For Heawood
conjecture . The case four-color theorem . The Klein
bottle is nonorientable and is outside this family.
A torus
therefore has worst-case map chromatic number A000934 ).
A set of regions requiring seven colors is shown above for an ordinary torus .
The above figure shows the relationship between the Heawood
graph and the 7-color torus coloring.
See also Chromatic Number ,
Four-Color Theorem ,
Heawood Conjecture ,
Heawood
Graph ,
Klein Bottle ,
Map
Coloring ,
Szilassi Polyhedron ,
Torus
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References Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. Cadwell,
J. H. Ch. 8 in Topics
in Recreational Mathematics. Gardner, M. "Mathematical Games: The Celebrated Four-Color
Map Problem of Topology." Sci. Amer. 203 , 218-222, Sep. 1960. Ringel,
G. Map
Color Theorem. Ringel, G. and
Youngs, J. W. T. "Solution of the Heawood Map-Coloring Problem."
Proc. Nat. Acad. Sci. USA 60 , 438-445, 1968. https://doi.org/10.1073/pnas.60.2.438 . Sloane,
N. J. A. Sequence A000934 /M3292
in "The On-Line Encyclopedia of Integer Sequences." Steinhaus,
H. Mathematical
Snapshots, 3rd ed. Wagon,
S. "Map Coloring on a Torus." §7.5 in Mathematica
in Action. Wells,
D. The
Penguin Dictionary of Curious and Interesting Numbers. Wells, D. The
Penguin Dictionary of Curious and Interesting Geometry. Referenced on Wolfram|Alpha Torus Coloring
Cite this as:
Weisstein, Eric W. "Torus Coloring." From
MathWorld https://mathworld.wolfram.com/TorusColoring.html
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, the maximum chromatic
number over maps with connected regions, where adjacent regions share a boundary
arc, is
is the floor function. For 
, sharpness of this bound was proved by Ringel and Youngs
(1968, pp. 438-439), resolving the Heawood
conjecture. The case 
is the four-color theorem. The Klein
bottle is nonorientable and is outside this family.
-holed torus
therefore has worst-case map chromatic number 
. 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 seven colors is shown above for an ordinary torus.
