Path: senator-bedfellow.mit.edu!bloom-beacon.mit.edu!news-peer.gip.net!news.gsl.net!gip.net!news.maxwell.syr.edu!sunqbc.risq.qc.ca!News.Dal.Ca!torn!watserv3.uwaterloo.ca!undergrad.math.uwaterloo.ca!neumann.uwaterloo.ca!alopez-o
From: [email protected] (Alex Lopez-Ortiz)
Newsgroups: sci.math,news.answers,sci.answers
Subject: sci.math FAQ: The Four Colour Theorem
Followup-To: sci.math
Date: 17 Feb 2000 22:52:04 GMT
Organization: University of Waterloo
Lines: 65
Approved: [email protected]
Expires: Sun, 1 Mar 1998 09:55:55
Message-ID: <[email protected]>
Reply-To: [email protected]
NNTP-Posting-Host: daisy.uwaterloo.ca
Summary: Part 17 of 31, New version
Originator: [email protected]
Originator: [email protected]
Xref: senator-bedfellow.mit.edu sci.math:347398 news.answers:177518 sci.answers:11226

Archive-name: sci-math-faq/fourcolour
Last-modified: February 20, 1998
Version: 7.5



                           The Four Colour Theorem

  Theorem 2 [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 3 [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 known proofs work by way of contradiction. The basic thrust of the
  proof is to assume that there are counterexamples, thus there must be
  minimal counterexamples in the sense that any subset of the graphic is
  four colourable. Then it is shown that all such minimal
  counterexamples must contain a subgraph from a set basic
  configurations.

  But it turns out that any one of those basic counterexamples can be
  replaced by something smaller, while preserving the need for five
  colours, thus contradicting minimality.

  The number of basic forms, or configurations has been reduced to 633.

  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. Available by anonymous ftp from
  ftp.math.gatech.edu, in directory /pub/users/thomas/fcdir/npfc.ps.

  The Four Color Theorem: Assault and Conquest T. Saaty and Paul Kainen.
  McGraw-Hill, 1977. Reprinted by Dover Publications 1986.
--
Alex Lopez-Ortiz                                         [email protected]
http://www.cs.unb.ca/~alopez-o                       Assistant Professor
Faculty of Computer Science                  University of New Brunswick