Skip to main content
SpringerLink
Log in
Book cover

A Course in Topological Combinatorics pp 97–143Cite as

Embedding and Mapping Problems

  • Chapter
  • First Online:

  • 3629 Accesses

Part of the book series: Universitext ((UTX))

Abstract

Embedding problems in discrete geometry lead to very challenging and interesting questions. A very basic question is whether a given graph G is planar, i.e., can be drawn in the plane such that edges do not cross. Kuratowski’s theorem, Theorem A.18, answers this question in terms of forbidden subgraphs. In this chapter we want to pursue an alternative characterization with methods from algebraic topology that reduce to simple linear algebra computations.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   49.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   64.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   84.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. Martin Aigner, Combinatorial search, Wiley-Teubner, 1988.

    Google Scholar 

  2. Noga Alon, Splitting necklaces, Adv. Math. 63 (1987), 247–253.

    Article  MATH  Google Scholar 

  3. Dan Archdeacon, The thrackle conjecture, http://www.emba.uvm.edu/~archdeac/problems/thrackle.htm. Accessed August 2012.

  4. Bradford H. Arnold, A topological proof of the fundamental theorem of algebra, Am. Math. Monthly 56 (1949), 465–466.

    Article  MATH  Google Scholar 

  5. Noga Alon and Joel H. Spencer, The probabilistic method, Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wiley & Sons, Inc., New York, 1992.

    Google Scholar 

  6. Imre Bárány, A short proof of Kneser’s conjecture, J. Combinatorial Theory, Ser. A 25 (1978), 325–326.

    Google Scholar 

  7. to3em______ , On a common generalization of Borsuk’s and Radon’s theorem, Acta Math. Acad. Sci. Hung. 34 (1979), 347–350.

    Google Scholar 

  8. M. R. Best, P. van Emde Boas, and H. W. jun. Lenstra, A sharpened version of the Aanderaa–Rosenberg conjecture, Afd. zuivere Wisk. ZW 300/74 (1974).

    Google Scholar 

  9. Thomas Bier, A remark on Alexander duality and the disjunct join, unpublished preprint, 7 pages, 1992.

    Google Scholar 

  10. R. H. Bing, Some aspects of the topology of 3-manifolds related to the Poincaré conjecture, Lectures on Modern Mathematics (T.L. Saaty, ed.), vol. II, Wiley, 1964, pp. 93–128.

    Google Scholar 

  11. Anders Björner, Shellable and Cohen–Macaulay partially ordered sets, Trans. AMS 260 (1980), no. 1, 159–183.

    Article  MATH  Google Scholar 

  12. to3em__________________________________________________________________________________________________ , Topological methods, Handbook of Combinatorics (R. Graham, M. Grötschel, and L. Lovász, eds.), North Holland, Amsterdam, 1994, pp. 1819–1872.

    Google Scholar 

  13. Eric Babson and Dmitry Kozlov, Proof of the Lovász conjecture, Annals of Mathematics (Second Series) 165 (2007), no. 3, 965–1007.

    Article  MathSciNet  MATH  Google Scholar 

  14. Bela Bollobas, Extremal graph theory, ch. 8. Complexity and Packing, Dover Publications, Mineola, New York (2004).

    Google Scholar 

  15. Glen E. Bredon, Introduction to compact transformation groups, Academic Press, New York-London (1972).

    Google Scholar 

  16. to3em___________ , Topology and geometry, Graduate Texts in Mathematics, vol. 139, Springer-Verlag, 1993.

    Google Scholar 

  17. Felix Breuer, Gauss codes and thrackles, Master’s thesis, Freie Universität Berlin, 2006.

    Google Scholar 

  18. Torsten Bronger, PP3 – Celestial Chart Generation, http://pp3.sourceforge.net/, 2003. Accessed March 2010.

  19. Imre Bárány, Senya B. Schlosman, and András Szűcs, On a topological generalization of a theorem of Tverberg, J. London Math. Soc. 23 (1981), no. 2, 158–164.

    Article  MathSciNet  MATH  Google Scholar 

  20. Bryan John Birch, On 3N points in a plane, Math. Proc. Cambridge Phil. Soc. 55 (1959), 289–293.

    Article  Google Scholar 

  21. Anders Björner and Michelle Wachs, On lexicographically shellable posets, Trans. AMS 277 (1983), no. 1, 323–341.

    Article  MATH  Google Scholar 

  22. Péter Csorba, Carsten Lange, Ingo Schurr, and Arnold Wassmer, Box complexes, neighborhood complexes, and the chromatic number, J. Comb. Theory, Ser. A 108 (2004), no. 1, 159–168.

    Google Scholar 

  23. Grant Cairns and Yury Nikolayevsky, Bounds for generalized thrackles, Discrete Comput. Geom. 23 (2000), no. 2, 191–206.

    Article  MathSciNet  MATH  Google Scholar 

  24. Reinhard Diestel, Graph theory, 3rd ed., Graduate Texts in Mathematics, vol. 173, Springer, New York, 2006.

    Google Scholar 

  25. Albrecht Dold, Simple proofs of some Borsuk–Ulam results, Contemp. Math. 19 (1983), 65–69.

    Article  MathSciNet  MATH  Google Scholar 

  26. Ky Fan, A generalization of Tucker’s combinatorial lemma with topological applications, Ann. Math. 56 (1952), no. 2, 431–437.

    Article  MATH  Google Scholar 

  27. Robert M. Freund and Michael J. Todd, A constructive proof of Tucker’s combinatorial lemma, J. Comb. Theory, Ser. A 30 (1981), 321–325.

    Google Scholar 

  28. David Gale, Neighboring vertices on a convex polyhedron, Linear Inequalities and Related Systems (H. W. Kuhn and A. W. Tucker, eds.), Annals of Math. Studies, vol. 38, Princeton University Press, 1956, pp. 255–263.

    Google Scholar 

  29. J. E. Green and Richard D. Ringeisen, Combinatorial drawings and thrackle surfaces, Graph Theory, Combinatorics, and Algorithms 1 (1992), no. 2, 999–1009.

    Google Scholar 

  30. Joshua E. Greene, A new short proof of Kneser’s conjecture, Amer. Math. Monthly 109 (2002), 918–920.

    Article  MathSciNet  MATH  Google Scholar 

  31. John Hopcroft and Robert Tarjan, Efficient planarity testing, J. of the Assoc. for Computing Machinery 21 (1974), no. 4, 549–568.

    Article  MathSciNet  MATH  Google Scholar 

  32. Thomas W. Hungerford, Algebra, Graduate Texts in Mathematics, vol. 73, Springer, 1974.

    Google Scholar 

  33. Elizabeth J. Jewell and Frank R. Abate (eds.), New Oxford American Dictionary, Oxford University Press, 2001.

    Google Scholar 

  34. Klaus Jänich, Topology, Springer-Verlag New York, 1984.

    Google Scholar 

  35. D.J. Kleitman and D.J. Kwiatkowski, Further results on the Aanderaa-Rosenberg conjecture, J. Comb. Theory, Ser. B 28 (1980), 85–95.

    Google Scholar 

  36. Martin Kneser, Aufgabe 360, Jahresbericht der Deutschen Mathematiker-Vereinigung 58 (1955), no. 2, 27.

    Google Scholar 

  37. Dmitry N. Kozlov, Chromatic numbers, morphism complexes, and Stiefel-Whitney characteristic classes, Geometric combinatorics (Ezra Miller et al., ed.), 13, Institute for Advanced Studies. IAS/Park City Mathematics, American Mathematical Society; Princeton, NJ, 2007, pp. 249–315.

    Google Scholar 

  38. Jeff Kahn, Michael Saks, and Dean Sturtevant, A topological approach to evasiveness, Combinatorica 4 (1984), no. 4, 297–306.

    Article  MathSciNet  MATH  Google Scholar 

  39. Mark de Longueville, Bier spheres and barycentric subdivision, J. Comb. Theory, Ser. A 105 (2004), 355–357.

    Google Scholar 

  40. László Lovász, Kneser’s conjecture, chromatic number and homotopy, J. Combinatorial Theory, Ser. A 25 (1978), 319–324.

    Google Scholar 

  41. László Lovász, János Pach, and Mario Szegedy, On Conway’s thrackle conjecture, Discrete Comput. Geom. 18 (1997), no. 4, 369–376.

    Article  MathSciNet  MATH  Google Scholar 

  42. William S. Massey, Algebraic topology: An introduction, Graduate Texts in Mathematics, vol. 56, Springer-Verlag, 1977.

    Google Scholar 

  43. to3em_____ , A basic course in algebraic topology, Graduate Texts in Mathematics, vol. 127, Springer Verlag, 1991.

    Google Scholar 

  44. Jiří Matoušek, A combinatorical proof of Kneser’s conjecture, Combinatorica 24 (2004), no. 1, 163–170.

    Article  MathSciNet  MATH  Google Scholar 

  45. J. Peter May, A concise course in algebraic topology, The University of Chicago Press, 1999.

    Google Scholar 

  46. Frédéric Meunier, A \({\mathbb{Z}}_{q}\) -Fan formula, preprint (2005), 14 pages.

    Google Scholar 

  47. James R. Munkres, Elements of algebraic topology, Addison-Wesley, Menlo Park, California, 1984.

    Google Scholar 

  48. Eric Charles Milner and Dominic J. A. Welsh, On the computational complexity of graph theoretical properties, Proc. 5th Br. comb. Conf. (Aberdeen 1975), 1976, pp. 471–487.

    Google Scholar 

  49. Jiří Matoušek and Günter M. Ziegler, Topological lower bounds for the chromatic number: a hierarchy., Jahresbericht der Deutschen Mathematiker-Vereinigung 106 (2004), no. 2, 71–90.

    MathSciNet  MATH  Google Scholar 

  50. Robert Oliver, Fixed-point sets of group actions on finite acyclic complexes, Comment. Math. Helv. 50 (1975), no. 50, 155–177.

    Article  MathSciNet  MATH  Google Scholar 

  51. Oystein Ore, The four color problem, Academic Press, New York, 1967.

    MATH  Google Scholar 

  52. Murad Özaydin, Equivariant maps for the symmetric group, unpublished preprint (1987), 17 pages, University of Wisconsin, Madison.

    Google Scholar 

  53. Igor Pak, The discrete square peg problem, http://arxiv.org/abs/0804.0657, 2008. Accessed June 2010

  54. George Pólya, On picture-writing, Am. Math. Mon. 63 (1956), 689–697.

    Article  MATH  Google Scholar 

  55. Timothy Prescott and Francis Edward Su, A constructive proof of Ky Fan’s generalization of Tucker’s lemma, J. Comb. Theory, Ser. A 111 (2005), no. 2, 257–265.

    Google Scholar 

  56. Michael J. Pelsmajer, Marcus Schaefer, and Daniel Štefankovič, Removing even crossings, J. Comb. Theory, Ser. B 97 (2007), 489–500.

    Google Scholar 

  57. Gian-Carlo Rota, The Lost Café, Los Alamos Science, Special Issue (1987), 23–32.

    Google Scholar 

  58. Ronald L. Rivest and Jean Vuillemin, On recognizing graph properties from adjacency matrices, Theor. Comput. Sci. 3 (1976), 371–384.

    Article  MathSciNet  Google Scholar 

  59. Jack Robertson and William Webb, Cake-cutting algorithms. Be fair if you can., A. K. Peters., 1998.

    Google Scholar 

  60. Thomas L. Saaty, Thirteen colorful variations on Guthrie’s Four–Color conjecture, Amer. Math. Monthly 79 (1972), no. 1, 2–43.

    Article  MathSciNet  MATH  Google Scholar 

  61. Karanbir S. Sarkaria, A one-dimensional Whitney trick and Kuratowski’s graph planarity criterion, Israel J. Math. 73 (1991), no. 1, 79–89.

    Article  MathSciNet  Google Scholar 

  62. to3em__ , Tverberg’s theorem via number fields, Israel J. Math. 79 (1992), no. 2-3, 317–320.

    Google Scholar 

  63. to3em______ , Tverberg partitions and Borsuk–Ulam theorems, Pacific J. Math. 196 (2000), 231–241.

    Google Scholar 

  64. Alexander Schrijver, Vertex critical subgraphs of Kneser graphs, Nieuw Arch. Wiskd. 26 (1978), no. III., 454–461.

    Google Scholar 

  65. Carsten Schultz, Graph colorings, spaces of edges and spaces of circuits, Advances in Math. 221 (2006), no. 6, 1733–1756.

    Article  Google Scholar 

  66. to3em_________________________________________________________________________________________ , On the \({\mathbb{Z}}_{2}\) -homotopy equivalence of \(\vert \mathcal{L}(G)\vert \) and |Hom (K 2 ,G) |, personal communication, 2010.

    Google Scholar 

  67. Daria Schymura, Über die Fragekomplexität von Mengen- und Grapheneigenschaften, Master’s thesis, Freie Universität Berlin, 2006.

    Google Scholar 

  68. K. Seetharaman, Pretzel production and quality control, Bakery Products: Science and Technology (Y. H. Hui, Harold Corke, Ingrid De Leyn, Wai-Kit Nip, and Nanna A. Cross, eds.), Blackwell, first ed., 2006, pp. 519–526.

    Google Scholar 

  69. Emanuel Sperner, Neuer Beweis für die Invarianz der Dimensionszahl und des Gebietes, Abhandlungen Hamburg 6 (1928), 265–272.

    Article  MATH  Google Scholar 

  70. Francis Edward Su and Forest W. Simmons, Consensus-halving via theorems of Borsuk–Ulam and Tucker, Math. Social Sci. 45 (2003), 15–25.

    Article  MathSciNet  MATH  Google Scholar 

  71. Gábor Simonyi and Gábor Tardos, Colorful subgraphs in Kneser-like graphs, European J. Combin. 28 (2007), no. 8, 2188–2200.

    Article  MathSciNet  MATH  Google Scholar 

  72. Francis Edward Su, Rental harmony: Sperner’s lemma in fair division, Am. Math. Monthly 10 (1999), 930–942.

    Google Scholar 

  73. Carsten Thomassen, The Jordan–Schoenflies theorem and the classification of surfaces, Amer. Math. Monthly 99 (1992), 116–130.

    Article  MathSciNet  MATH  Google Scholar 

  74. to3em___________________________________________________________________________________________________ , Embeddings and minors, Handbook of Combinatorics (R. Graham, M. Grötschel, and L. Lovász, eds.), North Holland, Amsterdam, 1994, pp. 301–349.

    Google Scholar 

  75. Robin Thomas, An update on the Four–Color theorem, Notices of the Amer. Math. Soc. 45 (1998), no. 7, 848–859.

    MATH  Google Scholar 

  76. William Thomas Tutte, Toward a theory of crossing numbers, J. Comb. Theory 8 (1970), 45–53.

    Article  Google Scholar 

  77. Helge Tverberg, A generalization of Radon’s theorem, J. London Math. Soc. 41 (1966), 123–128.

    Article  MathSciNet  MATH  Google Scholar 

  78. Egbert R. van Kampen, Komplexe in euklidischen Räumen, Abh. Math. Semin. Hamb. Univ. 9 (1932), 72–78, 152–153.

    Google Scholar 

  79. Alexey Yu. Volovikov, On a topological generalization of the Tverberg theorem, Math. Notes 59 (1996), no. 3, 324–326.

    Article  MathSciNet  Google Scholar 

  80. James W. Walker, From graphs to ortholattices and equivariant maps, J. Comb. Theory, Ser. B 35 (1983), 171–192.

    Google Scholar 

  81. Douglas B. West, Introduction to graph theory, Prentice-Hall, 2005.

    Google Scholar 

  82. Douglas R. Woodall, Thrackles and deadlock, Proc. Conf. Combinatorial Mathematics and Its Applications, Oxford 1969 (1971), 335–347.

    Google Scholar 

  83. to3em_________________________ , Dividing a cake fairly, J. Math. Anal. Appl. 78 (1980), 233–247.

    Google Scholar 

  84. E.C. Zeeman, On the dunce hat, Topology 2 (1963), 341–358.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Hochschule für Technik und Wirtschaft Berlin, University of Applied Sciences, Berlin, Germany

    Mark de Longueville

Authors
  1. Mark de Longueville
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer Science+Business Media New York

About this chapter

Cite this chapter

de Longueville, M. (2013). Embedding and Mapping Problems. In: A Course in Topological Combinatorics. Universitext. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-7910-0_4

Download citation

Publish with us

Policies and ethics

Search

Navigation