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The complexity of the four colour theorem

Part of: Computability and recursion theory Computational number theory

Published online by Cambridge University Press:  01 August 2010

Cristian S. Calude  and
Elena Calude
Cristian S. Calude
Affiliation:
Department of Computer Science, The University of Auckland, Private Bag 92019, Auckland, New Zealand (email: c.calude@auckland.ac.nz)http://www.cs.auckland.ac.nz/∼cristian
Elena Calude
Affiliation:
Institute of Information and Mathematical Sciences, Massey University at Albany, Private Bag 102-904, North Shore MSC, New Zealandhttp://www.massey.ac.nz/∼ecalude

Abstract

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The four colour theorem states that the vertices of every planar graph can be coloured with at most four colours so that no two adjacent vertices receive the same colour. This theorem is famous for many reasons, including the fact that its original 1977 proof includes a non-trivial computer verification. Recently, a formal proof of the theorem was obtained with the equational logic program Coq [G. Gonthier, ‘Formal proof–the four color theorem’, Notices of Amer. Math. Soc. 55 (2008) no. 11, 1382–1393]. In this paper we describe an implementation of the computational method introduced by C. S. Calude and co-workers [Evaluating the complexity of mathematical problems. Part 1’, Complex Systems 18 (2009) 267–285; A new measure of the difficulty of problems’, J. Mult. Valued Logic Soft Comput. 12 (2006) 285–307] to evaluate the complexity of the four colour theorem. Our method uses a Diophantine equational representation of the theorem. We show that the four colour theorem is in the complexity class ℭU,4. For comparison, the Riemann hypothesis is in class ℭU,3 while Fermat’s last theorem is in class ℭU,1.

MSC classification

Secondary: 03D15: Complexity of computation (including implicit computational complexity) 11Y16: Algorithms; complexity
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 13 , January 2010 , pp. 414 - 425
Copyright
Copyright © London Mathematical Society 2010

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