Advertisement

An Institution Isomorphism for Planar Graph Colouring

  • Giuseppe Scollo
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3051)

Abstract

Maximal planar graphs with vertex resp. edge colouring are naturally cast as (deceptively similar) institutions. One then tries to embody Tait’s equivalence algorithms into morphisms between them, and is lead to a partial redesign of those institutions. This paper aims at introducing a few pragmatic questions which arise in this case study, which also showcases the use of relational concepts and notations in the design of the subject institutions, and gives an outline of a solution to the problem of designing an isomorphism between them.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aigner, M.: Graphentheorie—Eine Eintwicklung aus dem 4-Farbenproblem. B.G. Teubner, Stuttgart (1984)Google Scholar
  2. 2.
    Appel, K., Haken, W.: Every planar map is four colorable. Part I. Discharging. Illinois J. Math. 21, 429–490 (1977)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Appel, K., Haken, W., Koch, J.: Every planar map is four colorable. Part II. Reducibility. Illinois J. Math. 21, 491–567 (1977)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Appel, K., Haken, W.: Every planar map is four colorable. Contemporary Math. 98 (1989)Google Scholar
  5. 5.
    Birkhoff, G.D.: The reducibility of maps. Amer. J. Math. 35, 114–128 (1913)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Cerioli, M.: Relationships between logical formalisms, Ph. D. Thesis, University of Genova (March 1993)Google Scholar
  7. 7.
    Ebbinghaus, H.-D.: Extended logics: the general framework. In: Barwise, J., Feferman, S. (eds.) Model-Theoretic Logics, pp. 25–76. Springer, Berlin (1985)Google Scholar
  8. 8.
    Fritsch, R., Fritsch, G.: The Four Colour Theorem. Springer, New York (1998)Google Scholar
  9. 9.
    Goguen, J.A., Burstall, R.M.: Introducing Institutions. In: Clarke, E., Kozen, D. (eds.) Logic of Programs 1983. LNCS, vol. 164, pp. 221–256. Springer, Heidelberg (1984)Google Scholar
  10. 10.
    Goguen, J.A., Burstall, R.M.: Institutions: Abstract model theory for specification and programming. J. Assoc. Comput. Mach. 39, 95–146 (1992)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Goguen, J.A., Roşu, G.: Institution Morphisms. Formal Aspects of Computing 13, 274–307 (2002), At URL: http://www.cs.ucsd.edu/users/goguen/pubs/ zbMATHCrossRefGoogle Scholar
  12. 12.
    Mac Lane, S.: Categories for the Working Mathematician. Springer, New York (1971)zbMATHGoogle Scholar
  13. 13.
    Matiyasevich, Y.: A Polynomial Related to Colourings of Triangulation of Sphere, July 4 (1997), At URL: http://logic.pdmi.ras.ru/~umat/Journal/Triangular/triang.htm
  14. 14.
    Matiyasevich, Y.: The Four Colour Theorem as a possible corollary of binomial summation. Theoretical Computer Science 257(1-2), 167–183 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Meseguer, J.: General Logics. In: Ebbinghaus, H.-D., et al. (eds.) Logic Colloquium 1987, pp. 275–329. North-Holland, Amsterdam (1989)Google Scholar
  16. 16.
    Robertson, N., Sanders, D.P., Seymour, P.D., Thomas, R.: The four colour theorem. J. Combin. Theory Ser. B 70, 2–44 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Saati, T.L.: Thirteen colorful variations on Guthrie’s four-color conjecture. American Mathematical Monthly 79(1), 2–43 (1972)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Salibra, A., Scollo, G.: Interpolation and compactness in categories of preinstitutions. Mathematical Structures in Computer Science 6, 261–286 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Scollo, G.: Graph colouring institutions. In: Berghammer, R., Möller, B. (eds.) 7th Seminar RelMiCS, 2nd Workshop Kleene Algebra, Christian-Albrechts- Universität zu Kiel, Bad Malente, Germany, May 12-17, pp. 288–297 (2003), At http://www.informatik.uni-kiel.de/~relmics7
  20. 20.
    Scollo, G.: Morphism-driven design of graph colouring institutions, RR 03/2003, University of Verona, Dipartimento di Informatica (March 2003)Google Scholar
  21. 21.
    Tait, P.G.: Note on a theorem in the geometry of position. Trans. Roy. Soc. Edinburgh 29, 657–660 (1880); printed in Scientific Papers 1, 408–411Google Scholar
  22. 22.
    Tait, P.G.: On Listing’s topology. Phil. Mag. V. Ser. 17, 30–46 (1884); printed in Scientific Papers 2, 85–98Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Giuseppe Scollo
    • 1
  1. 1.Dipartimento di InformaticaUniversità di VeronaVeronaItaly

Personalised recommendations