Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-23T13:36:26.429Z Has data issue: false hasContentIssue false

One More Probabilistic Reformulation of the Four Colour Conjecture

Published online by Cambridge University Press:  24 August 2010

Yu. MATIYASEVICH*
Affiliation:
St. Petersburg Department of Steklov Institute of Mathematics, 27 Fontanka, St. Petersburg 191023, Russia (e-mail: [email protected])

Abstract

The paper presents yet another way to reformulate the Four Colour Conjecture as a statement concerning conditional probabilities of certain events involving planar graphs.

Type
Paper
Copyright
Copyright © Cambridge University Press 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Appel, K. and Haken, W. (1977) Every planar map is four colorable I: Discharging. Illinois J. Math. 21 429490.Google Scholar
[2]Appel, K., Haken, W. and Koch, J. (1977) Every planar map is four colorable II: Reducibility. Illinois J. Math. 21 491567.Google Scholar
[3]Appel, K. and Haken, W. (1989) Every Planar Map is Four Colorable, Vol. 98 of Contemporary Mathematics, AMS, Providence, RI.Google Scholar
[4]Gonthier, G. (2008) Formal proof: The four-color theorem. Notices Amer. Math. Soc. 55 13821393.Google Scholar
[5]Goodall, A. J. (2004) Graph polynomials and the discrete Fourier transform. PhD thesis, University of Oxford.Google Scholar
[6]Goodall, A. J. (2006) Some new evaluations of the Tutte polynomial. J. Combin. Theory Ser. B 96 207224.Google Scholar
[7]Goodall, A. J. (2008) Parity, Eulerian subgraphs and the Tutte polynomial. J. Combin. Theory Ser. B 98 599628.Google Scholar
[8]Kauffman, L. H. (1990) Map coloring and the vector cross product. J. Combin. Theory Ser. B 48 145154.Google Scholar
[9]Kung, J. P. S. (2008) Coboundaries, flows, and Tutte polynomials of matrices. Ann. Combin. 12 183194.Google Scholar
[10]Matiyasevich, Yu. V. (2003) A probabilistic equivalent to the four-color conjecture (in Russian). Teor. Veroyatnost. i Primenen. 48 411416. Translation in Theory Probab. Appl. 48 (2004) 368–372.Google Scholar
[11]Matiyasevich, Yu. (2004) Some probabilistic restatements of the four color conjecture. J. Graph Theory 46 167179.CrossRefGoogle Scholar
[12]Penrose, R. (1971) Applications of negative dimensional tensors. In Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969), Academic Press, pp. 221244.Google Scholar
[13]Robertson, N., Sanders, D., Seymour, P. and Thomas, R. (1997) The four-colour theorem. J. Combin. Theory Ser. B 70 244.Google Scholar
[14]Scheim, D. E. (1974) The number of edge 3-colorings of a planar cubic graph as a permanent. Discrete Math. 8 377382.Google Scholar