Subject: sci.math FAQ: Four Colour Theorem 
Summary: Part 21 of many, New version,
Date: Fri, 17 Nov 1995 17:15:28 GMT
Nntp-Posting-Host: neumann.uwaterloo.ca


Archive-Name: sci-math-faq/fourcolour
Version: 6.2

The Four Colour Theorem

   Theorem [Four Colour Theorem] Every planar map with regions of simple
   borders can be coloured with 4 colours in such a way that no two
   regions sharing a non-zero length border have the same colour.



   An equivalent combinatorial interpretation is

   Theorem [Four Colour Theorem] Every loopless planar graph admits a
   vertex-colouring with at most four different colours.



   This theorem was proved with the aid of a computer in 1976. The proof
   shows that if aprox. 1,936 basic forms of maps can be coloured with
   four colours, then any given map can be coloured with four colours. A
   computer program coloured these basic forms. So far nobody has been
   able to prove it without using a computer. In principle it is possible
   to emulate the computer proof by hand computations.

   The number of basic forms, or configurations has been reduced to 633.
   The basic trust of the known proofs is to show that none of those
   basic configurations may appear in a minimal counterexample to the
   Four Colour Theorem because if one appeared, it could be replaced by
   something smaller, contradicting minimality.

   Then it is shown that all minimal counterexamples must contain one of
   the basic configurations.

   A recent simplification of the Four Colour Theorem proof, by
   Robertson, Sanders, Seymour and Thomas, has removed the cloud of doubt
   hanging over the complex original proof of Appel and Haken.

   The programs can now be obtained by ftp and easily checked over for
   correctness. Also the paper part of the proof is easier to verify.
   This new proof, if correct, should dispel all reasonable criticisms of
   the validity of the proof of this theorem.



   References

   K. Appel and W. Haken. Every planar map is four colorable. Bulletin of
   the American Mathematical Society, vol. 82, 1976 pp.711-712.



   K. Appel and W. Haken. Every planar map is four colorable. Illinois
   Journal of Mathematics, vol. 21, 1977, pp. 429-567.



   N. Robertson, D. Sanders, P. Seymour, R. Thomas The Four Colour
   Theorem Preprint, February 1994.



   The Four Color Theorem: Assault and Conquest T. Saaty and Paul Kainen.
   McGraw-Hill, 1977. Reprinted by Dover Publications 1986.




     _________________________________________________________________



    alopez-o@barrow.uwaterloo.ca
    Tue Apr 04 17:26:57 EDT 1995


