Fallacious proofs were given independently by Kempe (1879) and Tait (1880). Kempe's proof was accepted for a decade until Heawood showed an error using a map with 18
faces (although a map with nine faces suffices to show the fallacy). The Heawood
conjecture provided a very general assertion for map coloring, showing that in
a genus 0 space (including the
sphere or plane), four colors
suffice. Ringel and Youngs (1968) proved that for
genus 
, the upper bound provided by the
Heawood conjecture also give the necessary
number of colors, with the exception of the Klein bottle
(for which the Heawood formula gives seven, but the correct bound is six).
In December 2004, G. Gonthier of Microsoft Research in Cambridge, England (working with B. Werner of INRIA in France) announced that they had verified the Robertson
et al. proof by formulating the problem in the equational logic program Coq
and confirming the validity of each of its steps (Devlin 2005, Knight 2005).
J. Ferro (pers. comm., Nov. 8, 2005) has debunked a number of purported "short" proofs of the four-color theorem.
SEE ALSO: Chromatic Number,
Errera Graph,
Fritsch Graph,
Graph
Coloring,
Hadwiger-Nelson Problem,
Heawood Conjecture,
Kempe
Chain,
Kittell Graph,
Map
Coloring,
McGregor Map,
Six-Color
Theorem,
Soifer Graph,
Torus
Coloring
REFERENCES:
Appel, K. and Haken, W. "Every Planar Map is Four-Colorable, II: Reducibility."
Illinois J. Math. 21, 491-567, 1977.
Appel, K. and Haken, W. "The Solution of the Four-Color Map Problem." Sci.
Amer. 237, 108-121, 1977.
Appel, K. and Haken, W. "The Four Color Proof Suffices." Math. Intell. 8,
10-20 and 58, 1986.
Appel, K. and Haken, W. Every
Planar Map is Four-Colorable. Providence, RI: Amer. Math. Soc., 1989.
Appel, K.; Haken, W.; and Koch, J. "Every Planar Map is Four Colorable. I: Discharging."
Illinois J. Math. 21, 429-490, 1977.
Barnette, D. Map Coloring, Polyhedra, and the Four-Color Problem. Providence, RI: Math. Assoc.
Amer., 1983.
Birkhoff, G. D. "The Reducibility of Maps." Amer. Math. J. 35,
114-128, 1913.
Chartrand, G. "The Four Color Problem." §9.3 in Introductory
Graph Theory. New York: Dover, pp. 209-215, 1985.
Coxeter, H. S. M. "The Four-Color Map Problem, 1840-1890." Math.
Teach. 52, 283-289, 1959.
Devlin, K. "Devlin's Angle: Last Doubts Removed About the Proof of the Four
Color Theorem." Jan. 2005. https://www.maa.org/devlin/devlin_01_05.html.
Errera, A. Du colorage de cartes et de quelques questions d'analysis situs.
Ph.D. thesis. Paris: Gauthier-Villars, 1921.
Franklin, P. "Note on the Four Color Problem." J. Math. Phys. 16,
172-184, 1937-1938.
Franklin, P. The
Four-Color Problem. New York: Scripta Mathematica, Yeshiva College, 1941.
Gardner, M. "Mathematical Games: The Celebrated Four-Color Map Problem of Topology."
Sci. Amer. 203, 218-222, Sep. 1960.
Gardner, M. "The Four-Color Map Theorem." Ch. 10 in Martin Gardner's New Mathematical Diversions from Scientific American. New York:
Simon and Schuster, pp. 113-123, 1966.
Gardner, M. "Mathematical Games: Six Sensational Discoveries that Somehow or Another have Escaped Public Attention." Sci. Amer. 232, 127-132,
Apr. 1975.
Gardner, M. "Mathematical Games: On Tessellating the Plane with Convex Polygons."
Sci. Amer. 232, 112-117, Jul. 1975.
Gardner, M. The Last Recreations: Hydras, Eggs, and Other Mathematical Mystifications. New
York: Springer-Verlag, p. 86, 1997.
Gethner, E. and Springer, W. M. II. "How False Is Kempe's Proof of the
Four-Color Theorem?" Congr. Numer. 164, 159-175, 2003.
Harary, F. "The Four Color Conjecture." Graph
Theory. Reading, MA: Addison-Wesley, p. 5, 1994.
Heawood, P. J. "Map Colour Theorems." Quart. J. Math. 24,
332-338, 1890.
Heawood, P. J. "On the Four-Color Map Theorem." Quart. J. Pure
Math. 29, 270-285, 1898.
Hutchinson, J. P. and Wagon, S. "Kempe Revisited." Amer. Math.
Monthly 105, 170-174, 1998.
Kempe, A. B. "On the Geographical Problem of Four-Colors." Amer.
J. Math. 2, 193-200, 1879.
Kittell, I. "A Group of Operations on a Partially Colored Map." Bull.
Amer. Math. Soc. 41, 407-413, 1935.
Knight, W. "Computer Generates Verifiable Mathematics Proof." New Scientist
Breaking News. Apr. 19, 2005. https://www.newscientist.com/article.ns?id=dn7286.
Kraitchik, M. §8.4.2 in Mathematical
Recreations. New York: W. W. Norton, p. 211, 1942.
May, K. O. "The Origin of the Four-Color Conjecture." Isis 56,
346-348, 1965.
Morgenstern, C. and Shapiro, H. "Heuristics for Rapidly 4-Coloring Large Planar
Graphs." Algorithmica 6, 869-891, 1991.
Ore, Ø. The
Four-Color Problem. New York: Academic Press, 1967.
Ore, Ø. and Stemple, G. J. "Numerical Methods in the Four Color Problem." Recent Progress in Combinatorics (Ed. W. T. Tutte).
New York: Academic Press, 1969.
Pappas, T. "The Four-Color Map Problem: Topology Turns the Tables on Map Coloring." The
Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 152-153,
1989.
Ringel, G. and Youngs, J. W. T. "Solution of the Heawood Map-Coloring
Problem." Proc. Nat. Acad. Sci. USA 60, 438-445, 1968.
Robertson, N.; Sanders, D. P.; Seymour, P. D.; and Thomas, R. "A New Proof of the Four Colour Theorem." Electron. Res. Announc. Amer. Math. Soc. 2,
17-25, 1996.
Robertson, N.; Sanders, D. P.; and Thomas, R. "The Four-Color Theorem."
https://www.math.gatech.edu/~thomas/FC/fourcolor.html.
Saaty, T. L. and Kainen, P. C. The
Four-Color Problem: Assaults and Conquest. New York: Dover, 1986.
Skiena, S. Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading,
MA: Addison-Wesley, p. 210, 1990.
Steinhaus, H. Mathematical
Snapshots, 3rd ed. New York: Dover, pp. 274-275, 1999.
Tait, P. G. "Note on a Theorem in Geometry of Position." Trans.
Roy. Soc. Edinburgh 29, 657-660, 1880.
Thomas, R. "An Update on the Four-Color Theorem." Not. Amer. Math. Soc. 45,
858-857, 1998.
Weisstein, E. W. "Books about Four-Color Problem." https://www.ericweisstein.com/encyclopedias/books/Four-ColorProblem.html.
Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England:
Penguin Books, p. 57, 1986.
Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin,
pp. 81-82, 1991.
Wilson, R. Four Colors Suffice : How the Map Problem Was Solved. Princeton, NJ: Princeton
University Press, 2004.
CITE THIS AS:
Weisstein, Eric W. "Four-Color Theorem."
From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Four-ColorTheorem.html
case, and this
number can easily be reduced to five, but reducing the number of colors all the way
to four proved very difficult. This result was finally obtained by Appel and Haken
(1977), who constructed a computer-assisted proof that four colors were sufficient.
However, because part of the proof consisted of an exhaustive analysis of many discrete
cases by a computer, some mathematicians do not accept it. However, no flaws have
yet been found, so the proof appears valid. A shorter, independent proof was constructed
by Robertson et al. (1996; Thomas 1998).
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