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An Institution Isomorphism for Planar Graph Colouring

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Relational and Kleene-Algebraic Methods in Computer Science (RelMiCS 2003)

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

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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.

This research has been partially supported by MURST Grant prot. 2001017741 under project “Ragionamento su aggregati e numeri a supporto della programmazione e relative verifiche” at the DMI Department of the University of Catania.

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References

  1. Aigner, M.: Graphentheorie—Eine Eintwicklung aus dem 4-Farbenproblem. B.G. Teubner, Stuttgart (1984)

    Google Scholar 

  2. Appel, K., Haken, W.: Every planar map is four colorable. Part I. Discharging. Illinois J. Math. 21, 429–490 (1977)

    MATH  MathSciNet  Google Scholar 

  3. Appel, K., Haken, W., Koch, J.: Every planar map is four colorable. Part II. Reducibility. Illinois J. Math. 21, 491–567 (1977)

    MATH  MathSciNet  Google Scholar 

  4. Appel, K., Haken, W.: Every planar map is four colorable. Contemporary Math. 98 (1989)

    Google Scholar 

  5. Birkhoff, G.D.: The reducibility of maps. Amer. J. Math. 35, 114–128 (1913)

    Article  MathSciNet  Google Scholar 

  6. Cerioli, M.: Relationships between logical formalisms, Ph. D. Thesis, University of Genova (March 1993)

    Google Scholar 

  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. Fritsch, R., Fritsch, G.: The Four Colour Theorem. Springer, New York (1998)

    Google Scholar 

  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. Goguen, J.A., Burstall, R.M.: Institutions: Abstract model theory for specification and programming. J. Assoc. Comput. Mach. 39, 95–146 (1992)

    MATH  MathSciNet  Google Scholar 

  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/

    Article  MATH  Google Scholar 

  12. Mac Lane, S.: Categories for the Working Mathematician. Springer, New York (1971)

    MATH  Google Scholar 

  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. Matiyasevich, Y.: The Four Colour Theorem as a possible corollary of binomial summation. Theoretical Computer Science 257(1-2), 167–183 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  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. Robertson, N., Sanders, D.P., Seymour, P.D., Thomas, R.: The four colour theorem. J. Combin. Theory Ser. B 70, 2–44 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  17. Saati, T.L.: Thirteen colorful variations on Guthrie’s four-color conjecture. American Mathematical Monthly 79(1), 2–43 (1972)

    Article  MathSciNet  Google Scholar 

  18. Salibra, A., Scollo, G.: Interpolation and compactness in categories of preinstitutions. Mathematical Structures in Computer Science 6, 261–286 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  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. Scollo, G.: Morphism-driven design of graph colouring institutions, RR 03/2003, University of Verona, Dipartimento di Informatica (March 2003)

    Google Scholar 

  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, 408–411

    Google Scholar 

  22. Tait, P.G.: On Listing’s topology. Phil. Mag. V. Ser. 17, 30–46 (1884); printed in Scientific Papers 2, 85–98

    Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Informatica, Università di Verona, Ca’ Vignal, Strada Le Grazie 15, I-37134, Verona, Italy

    Giuseppe Scollo

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  1. Giuseppe Scollo
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Editors and Affiliations

  1. Institut für Informatik, Christian-Albrechts-Universität Kiel, Olshausenstraße 40, 24098, Kiel, Germany

    Rudolf Berghammer

  2. Institute of Computer Science, University of Augsburg,  

    Bernhard Möller

  3. Department of Computer Science, University of Sheffield, United Kingdom

    Georg Struth

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Scollo, G. (2004). An Institution Isomorphism for Planar Graph Colouring. In: Berghammer, R., Möller, B., Struth, G. (eds) Relational and Kleene-Algebraic Methods in Computer Science. RelMiCS 2003. Lecture Notes in Computer Science, vol 3051. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24771-5_22

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  • DOI: https://doi.org/10.1007/978-3-540-24771-5_22

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-22145-6

  • Online ISBN: 978-3-540-24771-5

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