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Graph Theory pp 115-133 | Cite as

To the Moon and Beyond

  • Ellen GethnerEmail author
Chapter
Part of the Problem Books in Mathematics book series (PBM)

Abstract

Are there nontrivial integer solutions to the equation xn + yn = zn for any integer n ≥ 2? How about for n > 2? Wait a minute! Is this a chapter about Fermat’s Last Theorem or about Graph Coloring? From one point of view, theorems such as The Four Color Theorem and Fermat’s Last Theorem are no more important than countless other problems in chromatic graph theory or in diophantine equations, respectively, and yet both problems are pervasive in science and culture.

Notes

Acknowledgements

I would like to express my appreciation to Thom Sulanke for our collaboration on the Earth-Moon problem, and for sharing his history and related correspondence, which has added another dimension of enjoyment to the adventure.

References

  1. 1.
    R. Alalqam, Heuristic methods applied to difficult graph theory problems. M.S. Thesis, University of Colorado Denver, 2012Google Scholar
  2. 2.
    M.O. Albertson, D.L. Boutin, E. Gethner, The thickness and chromatic number of r-inflated graphs. Discrete Math. 310(20), 2725–2734 (2010)MathSciNetCrossRefGoogle Scholar
  3. 3.
    B. Alspach, The wonderful Walecki construction. Bull. Inst. Combin. Appl. 52, 7–20 (2008)MathSciNetzbMATHGoogle Scholar
  4. 4.
    K. Appel, W. Haken, Every planar map is four colorable. Bull. Am. Math. Soc. 82(5), 711–712 (1976)MathSciNetCrossRefGoogle Scholar
  5. 5.
    K. Appel, W. Haken, A proof of the four color theorem. Discrete Math. 16(2), 179–180 (1976)MathSciNetCrossRefGoogle Scholar
  6. 6.
    K. Appel, W. Haken, Microfiche supplement to “Every planar map is four colorable. Part I and Part II”. Ill. J. Math. 21(3) (1977); by Appel and Haken; “II. Reducibility” (ibid. 21 (1977), no. 3, 491–567) by Appel, Haken and J. Koch. Illinois J. Math., 21(3):1–251. (microfiche supplement), 1977Google Scholar
  7. 7.
    K. Appel, W. Haken, J. Koch, Every planar map is four colorable. Part I and Part II. Ill. J. Math. 21(3), 491–567 (1977)MathSciNetzbMATHGoogle Scholar
  8. 8.
    D.L. Boutin, E. Gethner, T. Sulanke, Thickness-two graphs. I. New nine-critical graphs, permuted layer graphs, and Catlin’s graphs. J. Graph Theory 57(3), 198–214 (2008)MathSciNetCrossRefGoogle Scholar
  9. 9.
    J. Battle, F. Harary, Y. Kodama, Every planar graph with nine points has a nonplanar complement. Bull. Am. Math. Soc. 68, 569–571 (1962)MathSciNetCrossRefGoogle Scholar
  10. 10.
    G. Cornell, J.H. Silverman, G. Stevens (eds.), Modular Forms and Fermat’s Last Theorem (Springer, New York, 1997). Papers from the Instructional Conference on Number Theory and Arithmetic Geometry held at Boston University, Boston, MA, August 9–18, 1995zbMATHGoogle Scholar
  11. 11.
    J. Flower, Heuristic search methods for discovering thickness-N graphs. M.S. Thesis, University of Colorado Denver, 2017Google Scholar
  12. 12.
    M. Gardner, Mathematical games. Sci. Am. 242, 14–19 (1980)CrossRefGoogle Scholar
  13. 13.
    M. Gardner, The Last Recreations: Hydras, Eggs, and Other Mathematical Mystifications (Copernicus, New York, 1997)CrossRefGoogle Scholar
  14. 14.
    M.R. Garey, D.S. Johnson, Computers and Intractability; A Guide to the Theory of NP-Completeness (W. H. Freeman & Co., New York, 1990)zbMATHGoogle Scholar
  15. 15.
    E. Gethner, T. Sulanke, Thickness-two graphs. II. More new nine-critical graphs, independence ratio, cloned planar graphs, and singly and doubly outerplanar graphs. Graphs Combin. 25(2), 197–217 (2009)Google Scholar
  16. 16.
    P.J. Heawood, Map-colour theorem. Q. J. Pure Appl. Math. 24, 332–338 (1890)zbMATHGoogle Scholar
  17. 17.
    P. Hell, A. Kotzig, A. Rosa, Some results on the Oberwolfach problem (decomposition of complete graphs into isomorphic quadratic factors). Aequationes Math. 12, 1–5 (1975)MathSciNetCrossRefGoogle Scholar
  18. 18.
    N. Hartsfield, G. Ringel, Pearls in Graph Theory, rev. edn. (Dover Publications Inc., Mineola, 2003)zbMATHGoogle Scholar
  19. 19.
    B. Jackson, G. Ringel, Solution of Heawood’s empire problem in the plane. J. Reine Angew. Math. 347, 146–153 (1984)MathSciNetzbMATHGoogle Scholar
  20. 20.
    T.R. Jensen, B. Toft, Graph Coloring Problems. Wiley-Interscience Series in Discrete Mathematics and Optimization (Wiley, New York, 1995)Google Scholar
  21. 21.
    P.C. Kainen, Some recent results in topological graph theory, in Graphs and combinatorics (Proc. Capital Conf., George Washington Univ., Washington, D.C., 1973). Lecture Notes in Mathematics, vol. 406 (Springer, Berlin, 1974), pp. 76–108Google Scholar
  22. 22.
    E. Lucas, Recreations Mathematiques, four volumes (Gautheir–Villars, Paris, 1882–1894)Google Scholar
  23. 23.
    A. Mansfield, Determining the thickness of graphs is NP-hard. Math. Proc. Camb. Philos. Soc. 93(1), 9–23 (1983)MathSciNetCrossRefGoogle Scholar
  24. 24.
    C.St.J.A. Nash-Williams, Decomposition of finite graphs into forests. J. Lond. Math. Soc. 39, 12 (1964)MathSciNetCrossRefGoogle Scholar
  25. 25.
    G. Ringel, Färbungsprobleme auf Flächen und Graphen. Mathematische Monographien, vol. 2 (VEB Deutscher Verlag der Wissenschaften, Berlin, 1959)Google Scholar
  26. 26.
    N. Robertson, D.P. Sanders, P. Seymour, R. Thomas, A new proof of the four-colour theorem. Electron. Res. Announc. Am. Math. Soc. 2(1), 17–25 (electronic) (1996)Google Scholar
  27. 27.
    G. Ringel, J.W.T. Youngs, Solution of the Heawood map-coloring problem. Proc. Natl. Acad. Sci. U. S. A. 60, 438–445 (1968)MathSciNetCrossRefGoogle Scholar
  28. 28.
    R.B. Sheridan, The Rivals, 1st edn. (John Wilkie, London, 1775)Google Scholar
  29. 29.
  30. 30.
    T.W. Tillson, A Hamiltonian decomposition of \(K^{{ }^{\ast } }_{2m}\), 2m ≥ 8. J. Combin. Theory Ser. B 29(1), 68–74 (1980)Google Scholar
  31. 31.
    W.T. Tutte, The non-biplanar character of the complete 9-graph. Can. Math. Bull. 6, 319–330 (1963)MathSciNetCrossRefGoogle Scholar
  32. 32.
    E.J. Williams, Experimental designs balanced for the estimation of residual effects of treatments. Aust. J. Sci. Res. Ser. A 2, 149–168 (1949)MathSciNetGoogle Scholar
  33. 33.
    A. Wiles, Modular forms, elliptic curves, and Fermat’s last theorem, in Proceedings of the International Congress of Mathematicians, Zürich, 1994, vols. 1, 2 (Birkhäuser, Basel, 1995), pp. 243–245Google Scholar
  34. 34.
    R .Wilson, Four colors suffice: how the map problem was solved (Princeton University Press, Princeton, 2002)zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.University of Colorado DenverDenverUSA

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