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.
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References
Martin Aigner, Combinatorial search, Wiley-Teubner, 1988.
Noga Alon, Splitting necklaces, Adv. Math. 63 (1987), 247–253.
Dan Archdeacon, The thrackle conjecture, http://www.emba.uvm.edu/~archdeac/problems/thrackle.htm. Accessed August 2012.
Bradford H. Arnold, A topological proof of the fundamental theorem of algebra, Am. Math. Monthly 56 (1949), 465–466.
Noga Alon and Joel H. Spencer, The probabilistic method, Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wiley & Sons, Inc., New York, 1992.
Imre Bárány, A short proof of Kneser’s conjecture, J. Combinatorial Theory, Ser. A 25 (1978), 325–326.
to3em______ , On a common generalization of Borsuk’s and Radon’s theorem, Acta Math. Acad. Sci. Hung. 34 (1979), 347–350.
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).
Thomas Bier, A remark on Alexander duality and the disjunct join, unpublished preprint, 7 pages, 1992.
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.
Anders Björner, Shellable and Cohen–Macaulay partially ordered sets, Trans. AMS 260 (1980), no. 1, 159–183.
to3em__________________________________________________________________________________________________ , Topological methods, Handbook of Combinatorics (R. Graham, M. Grötschel, and L. Lovász, eds.), North Holland, Amsterdam, 1994, pp. 1819–1872.
Eric Babson and Dmitry Kozlov, Proof of the Lovász conjecture, Annals of Mathematics (Second Series) 165 (2007), no. 3, 965–1007.
Bela Bollobas, Extremal graph theory, ch. 8. Complexity and Packing, Dover Publications, Mineola, New York (2004).
Glen E. Bredon, Introduction to compact transformation groups, Academic Press, New York-London (1972).
to3em___________ , Topology and geometry, Graduate Texts in Mathematics, vol. 139, Springer-Verlag, 1993.
Felix Breuer, Gauss codes and thrackles, Master’s thesis, Freie Universität Berlin, 2006.
Torsten Bronger, PP3 – Celestial Chart Generation, http://pp3.sourceforge.net/, 2003. Accessed March 2010.
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.
Bryan John Birch, On 3N points in a plane, Math. Proc. Cambridge Phil. Soc. 55 (1959), 289–293.
Anders Björner and Michelle Wachs, On lexicographically shellable posets, Trans. AMS 277 (1983), no. 1, 323–341.
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.
Grant Cairns and Yury Nikolayevsky, Bounds for generalized thrackles, Discrete Comput. Geom. 23 (2000), no. 2, 191–206.
Reinhard Diestel, Graph theory, 3rd ed., Graduate Texts in Mathematics, vol. 173, Springer, New York, 2006.
Albrecht Dold, Simple proofs of some Borsuk–Ulam results, Contemp. Math. 19 (1983), 65–69.
Ky Fan, A generalization of Tucker’s combinatorial lemma with topological applications, Ann. Math. 56 (1952), no. 2, 431–437.
Robert M. Freund and Michael J. Todd, A constructive proof of Tucker’s combinatorial lemma, J. Comb. Theory, Ser. A 30 (1981), 321–325.
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.
J. E. Green and Richard D. Ringeisen, Combinatorial drawings and thrackle surfaces, Graph Theory, Combinatorics, and Algorithms 1 (1992), no. 2, 999–1009.
Joshua E. Greene, A new short proof of Kneser’s conjecture, Amer. Math. Monthly 109 (2002), 918–920.
John Hopcroft and Robert Tarjan, Efficient planarity testing, J. of the Assoc. for Computing Machinery 21 (1974), no. 4, 549–568.
Thomas W. Hungerford, Algebra, Graduate Texts in Mathematics, vol. 73, Springer, 1974.
Elizabeth J. Jewell and Frank R. Abate (eds.), New Oxford American Dictionary, Oxford University Press, 2001.
Klaus Jänich, Topology, Springer-Verlag New York, 1984.
D.J. Kleitman and D.J. Kwiatkowski, Further results on the Aanderaa-Rosenberg conjecture, J. Comb. Theory, Ser. B 28 (1980), 85–95.
Martin Kneser, Aufgabe 360, Jahresbericht der Deutschen Mathematiker-Vereinigung 58 (1955), no. 2, 27.
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.
Jeff Kahn, Michael Saks, and Dean Sturtevant, A topological approach to evasiveness, Combinatorica 4 (1984), no. 4, 297–306.
Mark de Longueville, Bier spheres and barycentric subdivision, J. Comb. Theory, Ser. A 105 (2004), 355–357.
László Lovász, Kneser’s conjecture, chromatic number and homotopy, J. Combinatorial Theory, Ser. A 25 (1978), 319–324.
László Lovász, János Pach, and Mario Szegedy, On Conway’s thrackle conjecture, Discrete Comput. Geom. 18 (1997), no. 4, 369–376.
William S. Massey, Algebraic topology: An introduction, Graduate Texts in Mathematics, vol. 56, Springer-Verlag, 1977.
to3em_____ , A basic course in algebraic topology, Graduate Texts in Mathematics, vol. 127, Springer Verlag, 1991.
Jiří Matoušek, A combinatorical proof of Kneser’s conjecture, Combinatorica 24 (2004), no. 1, 163–170.
J. Peter May, A concise course in algebraic topology, The University of Chicago Press, 1999.
Frédéric Meunier, A \({\mathbb{Z}}_{q}\) -Fan formula, preprint (2005), 14 pages.
James R. Munkres, Elements of algebraic topology, Addison-Wesley, Menlo Park, California, 1984.
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.
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.
Robert Oliver, Fixed-point sets of group actions on finite acyclic complexes, Comment. Math. Helv. 50 (1975), no. 50, 155–177.
Oystein Ore, The four color problem, Academic Press, New York, 1967.
Murad Özaydin, Equivariant maps for the symmetric group, unpublished preprint (1987), 17 pages, University of Wisconsin, Madison.
Igor Pak, The discrete square peg problem, http://arxiv.org/abs/0804.0657, 2008. Accessed June 2010
George Pólya, On picture-writing, Am. Math. Mon. 63 (1956), 689–697.
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.
Michael J. Pelsmajer, Marcus Schaefer, and Daniel Štefankovič, Removing even crossings, J. Comb. Theory, Ser. B 97 (2007), 489–500.
Gian-Carlo Rota, The Lost Café, Los Alamos Science, Special Issue (1987), 23–32.
Ronald L. Rivest and Jean Vuillemin, On recognizing graph properties from adjacency matrices, Theor. Comput. Sci. 3 (1976), 371–384.
Jack Robertson and William Webb, Cake-cutting algorithms. Be fair if you can., A. K. Peters., 1998.
Thomas L. Saaty, Thirteen colorful variations on Guthrie’s Four–Color conjecture, Amer. Math. Monthly 79 (1972), no. 1, 2–43.
Karanbir S. Sarkaria, A one-dimensional Whitney trick and Kuratowski’s graph planarity criterion, Israel J. Math. 73 (1991), no. 1, 79–89.
to3em__ , Tverberg’s theorem via number fields, Israel J. Math. 79 (1992), no. 2-3, 317–320.
to3em______ , Tverberg partitions and Borsuk–Ulam theorems, Pacific J. Math. 196 (2000), 231–241.
Alexander Schrijver, Vertex critical subgraphs of Kneser graphs, Nieuw Arch. Wiskd. 26 (1978), no. III., 454–461.
Carsten Schultz, Graph colorings, spaces of edges and spaces of circuits, Advances in Math. 221 (2006), no. 6, 1733–1756.
to3em_________________________________________________________________________________________ , On the \({\mathbb{Z}}_{2}\) -homotopy equivalence of \(\vert \mathcal{L}(G)\vert \) and |Hom (K 2 ,G) |, personal communication, 2010.
Daria Schymura, Über die Fragekomplexität von Mengen- und Grapheneigenschaften, Master’s thesis, Freie Universität Berlin, 2006.
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.
Emanuel Sperner, Neuer Beweis für die Invarianz der Dimensionszahl und des Gebietes, Abhandlungen Hamburg 6 (1928), 265–272.
Francis Edward Su and Forest W. Simmons, Consensus-halving via theorems of Borsuk–Ulam and Tucker, Math. Social Sci. 45 (2003), 15–25.
Gábor Simonyi and Gábor Tardos, Colorful subgraphs in Kneser-like graphs, European J. Combin. 28 (2007), no. 8, 2188–2200.
Francis Edward Su, Rental harmony: Sperner’s lemma in fair division, Am. Math. Monthly 10 (1999), 930–942.
Carsten Thomassen, The Jordan–Schoenflies theorem and the classification of surfaces, Amer. Math. Monthly 99 (1992), 116–130.
to3em___________________________________________________________________________________________________ , Embeddings and minors, Handbook of Combinatorics (R. Graham, M. Grötschel, and L. Lovász, eds.), North Holland, Amsterdam, 1994, pp. 301–349.
Robin Thomas, An update on the Four–Color theorem, Notices of the Amer. Math. Soc. 45 (1998), no. 7, 848–859.
William Thomas Tutte, Toward a theory of crossing numbers, J. Comb. Theory 8 (1970), 45–53.
Helge Tverberg, A generalization of Radon’s theorem, J. London Math. Soc. 41 (1966), 123–128.
Egbert R. van Kampen, Komplexe in euklidischen Räumen, Abh. Math. Semin. Hamb. Univ. 9 (1932), 72–78, 152–153.
Alexey Yu. Volovikov, On a topological generalization of the Tverberg theorem, Math. Notes 59 (1996), no. 3, 324–326.
James W. Walker, From graphs to ortholattices and equivariant maps, J. Comb. Theory, Ser. B 35 (1983), 171–192.
Douglas B. West, Introduction to graph theory, Prentice-Hall, 2005.
Douglas R. Woodall, Thrackles and deadlock, Proc. Conf. Combinatorial Mathematics and Its Applications, Oxford 1969 (1971), 335–347.
to3em_________________________ , Dividing a cake fairly, J. Math. Anal. Appl. 78 (1980), 233–247.
E.C. Zeeman, On the dunce hat, Topology 2 (1963), 341–358.
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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
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