Arnold's seminar
24.11.98

S.Duzhin
The four-colour theorem: a survey of recent results (after R.Thomas).

The talk is based on the article by R.Thomas in the August 1998 issue
of the "Notices of AMS" and the papers referred to therein.

There are two proofs of the 4 colour theorem (4CT):
   1. Appel and Haken (1977),
   2. N.Robertson, D.Sanders, P.Seymour and R.Thomas (1997).
Both proofs proceed by enumerating a certain complete set of basic
configurations and then showing that in each case a required colouring
can be obtained through a certain set of rules. 
The computer calculations needed in the second proof are by an order of
magnitude smaller than in the first proof.

The most interesting thing is that there are many reformulations of the 4
colour theorem belonging to different areas of mathematics. Among these, we
mention the following.

1. (L.Kauffman, 1990) 4CT is equivalent to a certain claim about the standard
vector product of several vectors in R^3.

2. (D.Bar-Natan, 1996) 4CT is equivalent to a certain property of the Lie
algebra weight systems for cubic graphs.

3. (Yu.Matiyasevich, 1997) 4CT is equivalent to a certain property of
divisibility by 7 for the values of some special linear functions with 
integer coefficients.