“Map Coloring”的概念、定义、习题解题方法及翻译-数学辞典

单词

Map Coloring

释义

Map Coloring

Given a map with Genus , Heawood showed in 1890 that the maximum number of colorsnecessary to color a map (the Chromatic Number) on an unbounded surface is

where

is the Floor Function,

is the Genus, and

is the EulerCharacteristic. This is the Heawood Conjecture. In 1968, for any orientable surface other than the Sphere(or equivalently, the Plane) and any nonorientable surface other than the Klein Bottle,

was shown tobe 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 andnonorientable surfaces with the exception of the Klein Bottle. For this case, the actual number of colors needed is six--one less than (Franklin 1934; Saaty 1986, p. 45).

surface
Klein Bottle	1	7	6
Möbius Strip		6	6
Plane	0	4	4
Projective Plane		6	6
Sphere	0	4	4
Torus	1	7	7

See also Chromatic Number, Four-Color Theorem, Heawood Conjecture, Six-Color Theorem,Torus Coloring

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.

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