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# sci.math FAQ: The Four Colour Theorem

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

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```                            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

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                                         alopez-o@unb.ca
http://www.cs.unb.ca/~alopez-o                       Assistant Professor
Faculty of Computer Science                  University of New Brunswick
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