The Four-Color Problem

  • Kenneth Appel
  • Wolfgang Haken


In 1976, the Four-Color Problem was solved: every map drawn on a sheet of paper can be colored with only four colors in such a way that countries sharing a common border receive different colors. This result was of interest to the mathematical community since many mathematicians had tried in vain for over a hundred years to prove this simple-sounding statement. Yet among mathematicians who were not aware of the developments leading to the proof, the outcome had rather dismaying aspects, for the proof made unprecedented use of computer computation; the correctness of the proof cannot be checked without the aid of a computer. Moreover, adding to the strangeness of the proof, some of the crucial ideas were perfected by computer experiments. One can never rule out the chance that a short proof of the Four-Color Theorem might some day be found, perhaps by the proverbial bright high-school student. But it is also conceivable that no such proof is possible. In this case a new and interesting type of theorem has appeared, one which has no proof of the traditional type.


Dual Graph Ring Size Good Configuration Discharge Procedure Ring Configuration 
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Suggestions for Further Reading


  1. Biggs, N.L., Lloyd, E.K. and Wilson, Robin J. Graph Theory 1736–1936. Clarendon Pr, Oxford, 1976.MATHGoogle Scholar


  1. Appel, Kenneth and Haken, Wolfgang. Every planar map is four colorable, part I: discharging. Illinois J. of Mathematics 21 (1977) 429–490.MATHMathSciNetGoogle Scholar
  2. Appel, Kenneth, Haken, Wolfgang, and Koch, John. Every planar map is four colorable, part II: reducibility. Illinois J. of Mathematics 21 (1977) 491–567.MATHMathSciNetGoogle Scholar
  3. Harary, Frank. Graph Theory. Addison-Wesley, Reading, 1969.Google Scholar
  4. Ore, Oystein. The Four Color Problem. Academic Pr, New York, 1967.MATHGoogle Scholar
  5. Saaty, Thomas L. Thirteen colorful variations on Guthrie’s four color conjecture. American Mathematical Monthly 79 (1972) 2–43.CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Conference Board of the Mathematical Sciences 1978

Authors and Affiliations

  • Kenneth Appel
  • Wolfgang Haken

There are no affiliations available

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