Heawood Conjecture
The bound for the number of colors which are Sufficient for Map Coloring on a surface ofGenus 
,
is the best possible, where
is the Floor Function. 
is called the Chromatic Number, and thefirst few values for 
, 1, ... are 4, 7, 8, 9, 10, 11, 12, 12, 13, 13, 14, ... (Sloane's A000934).The fact that is also Necessary was proved by Ringel and Youngs (1968) with two exceptions:the Sphere (Plane), and the Klein Bottle (for which the Heawood Formula gives seven, but thecorrect bound is six). When the Four-Color Theorem was proved in 1976, the Klein Bottle was left as the onlyexception. The four most difficult cases to prove were 
, 83, 158, and 257.
References
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.
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