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The complexity of the four colour theorem

  • Cristian S. Calude (a1) and Elena Calude (a2)
Abstract

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.

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References
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[1] Appel, K. and Haken, W., ‘Every planar map is four-colorable, II: reducibility’, Illinois J. Math. 21 (1977) 491567.
[2] Appel, K., Haken, W. and Koch, J., ‘Every planar map is four colourable, I: discharging’, Illinois J. Math. 21 (1977) 429490.
[3] Calude, A. S., ‘The journey of the four colour theorem through time’, NZ Math. Magazine 38 (2001) no. 3, 2735.
[4] Calude, C. S., Information and randomness: an algorithmic perspective, 2nd edn (Springer, Berlin, 2002) revised and extended.
[5] Calude, C. S. and Calude, E., ‘Evaluating the complexity of mathematical problems. Part 1’, Complex Systems 18 (2009) 267285.
[6] Calude, C. S. and Calude, E., ‘Evaluating the complexity of mathematical problems. Part 2’, Complex Systems 18 (2010) 387401.
[7] Calude, C. S., Calude, E. and Dinneen, M. J., ‘A new measure of the difficulty of problems’, J. Mult. Valued Logic Soft Comput. 12 (2006) 285307.
[8] Calude, C. S., Calude, E. and Marcus, S., ‘Passages of proof’, Bull. EATCS 84 (2004) 167188.
[9] Calude, C. S., Calude, E. and Marcus, S., ‘Proving and programming’, Randomness & complexity, from Leibniz to Chaitin (ed. Calude, C. S.; World Scientific, Singapore, 2007) 310321.
[10] Calude, C. S., Dinneen, M. J. and Shu, C.-K., ‘Computing a glimpse of randomness’, Experiment. Math. 11 (2002) no. 2, 369378.
[11] Calude, E., ‘The complexity of the Goldbach’s conjecture and Riemann’s hypothesis’, CDMTCS Research Report 369, 2009, 14 pp.
[12] Chaitin, G. J., Algorithmic information theory (Cambridge University Press, Cambridge, 1987) third printing 1990.
[13] Davis, M., Matijasevič, Y. V. and Robinson, J., ‘Hilbert’s tenth problem. Diophantine equations: positive aspects of a negative solution’, Mathematical developments arising from Hilbert problems (ed. Browder, F. E.; American Mathematical Society, Providence, RI, 1976) 323378.
[14] Gonthier, G., ‘Formal proof—the four color theorem’, Notices Amer. Math. Soc. 55 (2008) no. 11, 13821393.
[15] Marcus, S., ‘Mathematics through the glasses of Hjelmslev’s semiotics’, Semiotica 1451/4 (2003) 235246.
[16] Robertson, N., Sanders, D. P., Seymour, P. and Thomas, R., ‘The four color theorem’, http://www.math.gatech.edu/∼thomas/FC/fourcolor.html (accessed 30 November 2008).
[17] Wilson, R., Four colours suffice (Penguin, London, 2002).
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LMS Journal of Computation and Mathematics
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  • EISSN: 1461-1570
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