Map Coloring

Given a map with genus , Heawood showed in 1890 that the maximum number  of colors necessary to color a map (the chromatic number) on an unbounded surface is

			(1)
			(2)

where  is the floor function,  is the genus, and  is the Euler characteristic. This is the Heawood conjecture. In 1968, for any unbounded orientable surface other than the sphere (or equivalently, the plane) and any nonorientable surface other than the Klein bottle,  was shown to be not merely a maximum, but the actual number needed (Ringel and Youngs 1968).

When the four-color theorem was proven, the Heawood formula was shown to hold also for all orientable and nonorientable unbounded surfaces with the exception of the Klein bottle. For the Klein bottle only, the actual number of colors  needed is six--one less than  (Franklin 1934; Saaty 1986, p. 45). The Möbius strip, which is a bounded surface, also requires 6 colors, while blind application of the Heawood formula (which is not applicable in this case) gives 7.

surface
Klein bottle	0	7	6
Möbius strip	0	7	6
plane	2	4	4
projective plane	1	6	6
sphere	2	4	4
torus	0	7	7

REFERENCES

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, pp. 237-238, 1987.

Barnette, D. Map Coloring, Polyhedra, and the Four-Color Problem. Washington, DC: Math. Assoc. Amer., 1983.

Franklin, P. "A Six Colour Problem." J. Math. Phys. 13, 363-369, 1934.

Franklin, P. The Four-Color Problem. New York: Scripta Mathematica, Yeshiva College, 1941.

Ore, Ø. The Four-Color Problem. New York: Academic Press, 1967.

Ringel, G. and Youngs, J. W. T. "Solution of the Heawood Map-Coloring Problem." Proc. Nat. Acad. Sci. USA 60, 438-445, 1968.

Saaty, T. L. and Kainen, P. C. The Four-Color Problem: Assaults and Conquest. New York: Dover, 1986.