Abstract
Based on an inductive definition of triangulations, a theory of undirected planar graphs is developed in Isabelle/HOL. The proof of the 5 colour theorem is discussed in some detail, emphasizing the readability of the computer assisted proofs.
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References
K. Appel and W. Haken. Every planar map is four colourable. Bulletin of the American mathematical Society, 82:711–112, 1977.
K. Appel and W. Haken. Every planar map is four colourable. I: Discharging. Illinois J. math., 21:429–490, 1977.
K. Appel and W. Haken. Every planar map is four colourable. II: Reducibility. Illinois J. math., 21:491–567, 1977.
K. Appel and W. Haken. Every planar map is four colourable, volume 98. Contemporary Mathematics, 1989.
G. D. Birkhoff. The reducibility of maps. Amer. J. Math., 35:114–128, 1913.
C.-T. Chou. A Formal Theory of Undirected Graphs in Higher-Order Logic. In T. Melham and J. Camilleri, editors, Higher Order Logic Theorem Proving and Its Applications, volume 859 of LNCS, pages 158–176, 1994.
R. Diestel. Graph Theory. Graduate Texts in Mathematics. Springer, 2000. Electronic Version: http://www.math.uni-hamburg.de/home/diestel/books/graph.theory/.
T. Emden-Weinert, S. Hougardy, B. Kreuter, H. Prömel, and A. Steger. Einführung in Graphen und Algorithmen, 1996. http://www.informatik.hu-berlin.de/Institut/struktur/algorithmen/ga/.
H. Heesch. Untersuchungen zum Vierfarbenproblem. Number 810/a/b in Hochschulskriptum. Bibliographisches Institut, Mannheim, 1969.
T. Nipkow, L. Paulson, and M. Wenzel. Isabelle/HOL — A Proof Assistant for Higher-Order Logic, volume 2283 of LNCS. Springer, 2002. To appear.
N. Robertson, D. Sanders, P. Seymour, and R. Thomas. The four colour theorem. http://www.math.gatech.edu/~thomas/FC/fourcolor.html, 1995.
N. Robertson, D. Sanders, P. Seymour, and R. Thomas. A new proof of the four colour theorem. Electron. Res. Announce. Amer. Math Soc., 2(1):17–25, 1996.
N. Robertson, D. Sanders, P. Seymour, and R. Thomas. The four colour theorem. J. Combin. Theory Ser. B, 70:2–44, 1997.
M. Wenzel. Isabelle/Isar— A Versatile Environment for Human-Readable Formal Proof Documents. PhD thesis, Institut für Informatik, Technische Universität München, 2002. http://tumb1.biblio.tu-muenchen.de/publ/diss/in/2002/wenzel.htm.
D. West. Introduction to Graph Theory. Prentice Hall, New York, 1996.
F. Wiedijk. The Four Color Theorem Project. http://www.cs.kun.nl/~freek/4ct/, 2000.
W. Wong. A simple graph theory and its application in railway signalling. In M. Archer, J. Joyce, K. Levitt, and P. Windley, editors, Proc. 1991 International Workshop on the HOL Theorem Proving System and its Applications, pages 395–409. IEEE Computer Society Press, 1992.
M. Yamamoto, S.-y. Nishizaha, M. Hagiya, and Y. Toda. Formalization of Planar Graphs. In E. Schubert, P. Windley, and J. Alves-Foss, editors, Higher Order Logic Theorem Proving and Its Applications, volume 971 of LNCS, pages 369–384, 1995.
M. Yamamoto, K. Takahashi, M. Hagiya, S.-y. Nishizaki, and T. Tamai. Formalization of Graph Search Algorithms and Its Applications. In J. Grundy and M. Newey, editors, Theorem Proving in Higher Order Logics, volume 1479 of LNCS, pages 479–496, 1998.
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Bauer, G., Nipkow, T. (2002). The 5 Colour Theorem in Isabelle/Isar. In: Carreño, V.A., Muñoz, C.A., Tahar, S. (eds) Theorem Proving in Higher Order Logics. TPHOLs 2002. Lecture Notes in Computer Science, vol 2410. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45685-6_6
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DOI: https://doi.org/10.1007/3-540-45685-6_6
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