Heawood Conjecture

The bound for the number of colors which are sufficient for map coloring on a surface of genus ,

is the best possible, where  is the floor function.  is called the chromatic number, and the first few values for , 1, ... are 4, 7, 8, 9, 10, 11, 12, 12, 13, 13, 14, ... (OEIS A000934).

The fact that  is also necessary was proved by Ringel and Youngs (1968) with two exceptions: the sphere (and plane), and the Klein bottle. When the four-color theorem was proved in 1976, the Klein bottle was left as the only exception, in that the Heawood formula gives seven, but the correct bound is six (as demonstrated by the Franklin graph). The four most difficult cases to prove in the Heawood conjecture were , 83, 158, and 257.

REFERENCES

Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, p. 244, 1976.

Franklin, P. "A Six Color Problem." J. Math. Phys. 13, 363-379, 1934.

Heawood, P. J. "Map Colour Theorem." Quart. J. Math. 24, 332-338, 1890.

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. Sequence A000934/M3292 in "The On-Line Encyclopedia of Integer Sequences."Wagon, S. "Map Coloring on a Torus." §7.5 in Mathematica in Action. New York: W. H. Freeman, pp. 232-237, 1991.