Torus Coloring
The number of colors Sufficient for Map Coloring on a surface of Genus 
isgiven by the Heawood Conjecture,
where
is the Floor Function. The fact that 
(which is called the Chromatic Number) isalso Necessary was proved by Ringel and Youngs (1968) with two exceptions: the Sphere (which requires thesame 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, 8, 9, 10, 11, 12, 12, 13, 13, 14, 15,15, 16, ... (Sloane's A000934).See also Chromatic Number, Four-Color Theorem, Heawood Conjecture, Klein Bottle, Map Coloring
References
Gardner, M. ``Mathematical Games: The Celebrated Four-Color Map Problem of Topology.'' Sci. Amer. 203, 218-222, Sep. 1960. Ringel, G. Map Color Theorem. New York: Springer-Verlag, 1974. Ringel, G. and Youngs, J. W. T. ``Solution of the Heawood Map-Coloring Problem.'' Proc. Nat. Acad. Sci. USA 60, 438-445, 1968. Sloane, N. J. A. SequenceA000934/M3292in ``An On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995. Wagon, S. ``Map Coloring on a Torus.'' §7.5 in Mathematica in Action. New York: W. H. Freeman, pp. 232-237, 1991.
- Whitehead Link
- Whitehead Manifold
- Whitehead's Theorem
- Whitney-Graustein Theorem
- Whitney-Mikhlin Extension Constants
- Whitney Singularity
- Whitney Sum
- Whitney Umbrella
- Whittaker Differential Equation
- Whittaker Function
- Whole Number
- Width (Partial Order)
- Width (Size)
- Wiedersehen Manifold
- Wieferich Prime
- Wielandt's Theorem
- Wiener Filter
- Wiener Function
- Wiener-Khintchine Theorem
- Wiener Measure
- Wiener Space
- Wigner 3j-Symbol
- Wigner 6j-Symbol
- Wigner 9j-Symbol
- Wigner-Eckart Theorem