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2013 | OriginalPaper | Buchkapitel

The Rectilinear Crossing Number of K n : Closing in (or Are We?)

verfasst von : Bernardo M. Ábrego, Silvia Fernández-Merchant, Gelasio Salazar

Erschienen in: Thirty Essays on Geometric Graph Theory

Verlag: Springer New York

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Abstract

The calculation of the rectilinear crossing number of complete graphs is an important open problem in combinatorial geometry, with important and fruitful connections to other classical problems. Our aim in this chapter is to survey the body of knowledge around this parameter.
Mathematics Subject Classification (2010): 52C30, 52C10, 52C45, 05C62, 68R10, 60D05, 52A22

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Fußnoten
1
Aichholzer, personal communication.
 
Literatur
1.
Zurück zum Zitat B.M. Ábrego, J. Balogh, S. Fernández-Merchant, J. Leaños, G. Salazar, An extended lower bound on the number of ( ≤  k)-edges to generalized configurations of points and the pseudolinear crossing number of K n . J. Comb. Theor. Ser. A 115, 1257–1264 (2008)MATHCrossRef B.M. Ábrego, J. Balogh, S. Fernández-Merchant, J. Leaños, G. Salazar, An extended lower bound on the number of ( ≤ ​ k)-edges to generalized configurations of points and the pseudolinear crossing number of K n . J. Comb. Theor. Ser. A 115, 1257–1264 (2008)MATHCrossRef
2.
Zurück zum Zitat B.M. Ábrego, M. Cetina, S. Fernández-Merchant, J. Leaños, G. Salazar, 3-Symmetric and 3-decomposable drawings of K n . Discr. Appl. Math. 158, 1240–1258 (2010)MATHCrossRef B.M. Ábrego, M. Cetina, S. Fernández-Merchant, J. Leaños, G. Salazar, 3-Symmetric and 3-decomposable drawings of K n . Discr. Appl. Math. 158, 1240–1258 (2010)MATHCrossRef
3.
Zurück zum Zitat B.M. Ábrego, M. Cetina, S. Fernández-Merchant, J. Leaños, G. Salazar, On ( ≤  k)-edges, crossings, and halving lines of geometric drawings of K n . Discrete Comput. Geom. 48, 192–215 (2012)MathSciNetMATHCrossRef B.M. Ábrego, M. Cetina, S. Fernández-Merchant, J. Leaños, G. Salazar, On ( ≤ ​ k)-edges, crossings, and halving lines of geometric drawings of K n . Discrete Comput. Geom. 48, 192–215 (2012)MathSciNetMATHCrossRef
4.
Zurück zum Zitat B.M. Ábrego, S. Fernández-Merchant, A lower bound for the rectilinear crossing number. Graphs Comb. 21, 293–300 (2005)MATHCrossRef B.M. Ábrego, S. Fernández-Merchant, A lower bound for the rectilinear crossing number. Graphs Comb. 21, 293–300 (2005)MATHCrossRef
5.
Zurück zum Zitat B.M. Ábrego, S. Fernández-Merchant, Geometric drawings of K n with few crossings. J. Comb. Theor. Ser. A 114, 373–379 (2007)MATHCrossRef B.M. Ábrego, S. Fernández-Merchant, Geometric drawings of K n with few crossings. J. Comb. Theor. Ser. A 114, 373–379 (2007)MATHCrossRef
6.
Zurück zum Zitat B.M. Ábrego, S. Fernández-Merchant, J. Leaños, G. Salazar, The maximum number of halving lines and the rectilinear crossing number of K n for n ≤ 27. Electron. Notes Discr. Math. 30, 261–266 (2008)CrossRef B.M. Ábrego, S. Fernández-Merchant, J. Leaños, G. Salazar, The maximum number of halving lines and the rectilinear crossing number of K n for n ≤ 27. Electron. Notes Discr. Math. 30, 261–266 (2008)CrossRef
7.
Zurück zum Zitat B.M. Ábrego, S. Fernández-Merchant, J. Leaños, G. Salazar, A central approach to bound the number of crossings in a generalized configuration. Electron. Notes Discr. Math. 30, 273–278 (2008)CrossRef B.M. Ábrego, S. Fernández-Merchant, J. Leaños, G. Salazar, A central approach to bound the number of crossings in a generalized configuration. Electron. Notes Discr. Math. 30, 273–278 (2008)CrossRef
8.
Zurück zum Zitat O. Aichholzer, F. Aurehnammer, H. Krasser, Enumerating order types for small point sets with applications. Order 19, 265–281 (2002)MathSciNetMATHCrossRef O. Aichholzer, F. Aurehnammer, H. Krasser, Enumerating order types for small point sets with applications. Order 19, 265–281 (2002)MathSciNetMATHCrossRef
9.
10.
Zurück zum Zitat O. Aichholzer, J. García, D. Orden, P. Ramos, New lower bounds for the number of ( ≤ k)-edges and the rectilinear crossing number of K n . Discr. Comput. Geom. 38, 1–14 (2007)MATHCrossRef O. Aichholzer, J. García, D. Orden, P. Ramos, New lower bounds for the number of ( ≤ k)-edges and the rectilinear crossing number of K n . Discr. Comput. Geom. 38, 1–14 (2007)MATHCrossRef
11.
Zurück zum Zitat O. Aichholzer, J. García, D. Orden, P. Ramos, New results on lower bounds for the number of ( ≤  k)-facets. Eur. J. Comb. 30, 1568–1574 (2009)MATHCrossRef O. Aichholzer, J. García, D. Orden, P. Ramos, New results on lower bounds for the number of ( ≤ ​ k)-facets. Eur. J. Comb. 30, 1568–1574 (2009)MATHCrossRef
12.
Zurück zum Zitat O. Aichholzer, H. Krasser, The point set order type data base: a collection of applications and results. in Proceedings of the 13th Annual Canadian Conference on Computational Geometry CCCG 2001, Waterloo, Ontario, 2001, pp. 17–20 O. Aichholzer, H. Krasser, The point set order type data base: a collection of applications and results. in Proceedings of the 13th Annual Canadian Conference on Computational Geometry CCCG 2001, Waterloo, Ontario, 2001, pp. 17–20
13.
Zurück zum Zitat O. Aichholzer, H. Krasser, Abstract order type extension and new results on the rectilinear crossing number. Comput. Geom. Theor. Appl. 36, 2–15 (2006)MathSciNetCrossRef O. Aichholzer, H. Krasser, Abstract order type extension and new results on the rectilinear crossing number. Comput. Geom. Theor. Appl. 36, 2–15 (2006)MathSciNetCrossRef
14.
Zurück zum Zitat J. Balogh, G. Salazar, k-Sets, convex quadrilaterals, and the rectilinear crossing number of K n . Discr. Comput. Geom. 35, 671–690 (2006)MathSciNetMATHCrossRef J. Balogh, G. Salazar, k-Sets, convex quadrilaterals, and the rectilinear crossing number of K n . Discr. Comput. Geom. 35, 671–690 (2006)MathSciNetMATHCrossRef
15.
Zurück zum Zitat W. Blaschke, Über Affine Geometrie XI: Lösung des “Vierpunktproblems” von Sylvester aus der Theorie der geometrischen Wahrscheinlichkeiten. Leipziger Berichte 69, 436–453 (1917) W. Blaschke, Über Affine Geometrie XI: Lösung des “Vierpunktproblems” von Sylvester aus der Theorie der geometrischen Wahrscheinlichkeiten. Leipziger Berichte 69, 436–453 (1917)
16.
Zurück zum Zitat P. Brass, W. Moser, J. Pach, Research Problems in Discrete Geometry (Springer, New York, 2005)MATH P. Brass, W. Moser, J. Pach, Research Problems in Discrete Geometry (Springer, New York, 2005)MATH
17.
Zurück zum Zitat A. Brodsky, S. Durocher, E. Gethner, The rectilinear crossing number of K 10 is 62. Electron. J. Comb. 8, R23 (2001)MathSciNet A. Brodsky, S. Durocher, E. Gethner, The rectilinear crossing number of K 10 is 62. Electron. J. Comb. 8, R23 (2001)MathSciNet
18.
Zurück zum Zitat A. Brodsky, S. Durocher, E. Gethner, Toward the rectilinear crossing number of K n : new drawings, upper bounds, and asymptotics. Discr. Math. 262, 59–77 (2003)MathSciNetMATHCrossRef A. Brodsky, S. Durocher, E. Gethner, Toward the rectilinear crossing number of K n : new drawings, upper bounds, and asymptotics. Discr. Math. 262, 59–77 (2003)MathSciNetMATHCrossRef
19.
Zurück zum Zitat M. Cetina, C. Hernández-Vélez, J. Leaños, C. Villalobos, Point sets that minimize ( ≤  k)-edges, 3-decomposable drawings, and the rectilinear crossing number of K 30. Discr. Math. 311, 1646–1657 (2011)MATHCrossRef M. Cetina, C. Hernández-Vélez, J. Leaños, C. Villalobos, Point sets that minimize ( ≤ ​ k)-edges, 3-decomposable drawings, and the rectilinear crossing number of K 30. Discr. Math. 311, 1646–1657 (2011)MATHCrossRef
20.
Zurück zum Zitat M.W. Crofton, Probability, in Encyclopedia Britannica, 9th edn. Philadelphia, PA: J. M. Stoddart 19, 768–788 (1885) M.W. Crofton, Probability, in Encyclopedia Britannica, 9th edn. Philadelphia, PA: J. M. Stoddart 19, 768–788 (1885)
21.
Zurück zum Zitat R.B. Eggleton, Ph.D. thesis, University of Calgary, 1973 R.B. Eggleton, Ph.D. thesis, University of Calgary, 1973
22.
Zurück zum Zitat R.K. Guy, Crossing numbers of graphs. Graph theory and applications, in Proceeding Conference, Western Michigan University, Kalamazoo, Mich., dedicated to the memory of J. W. T. Youngs), Lecture Notes in Math., Springer, Berlin, 303, 111–124 (1972) R.K. Guy, Crossing numbers of graphs. Graph theory and applications, in Proceeding Conference, Western Michigan University, Kalamazoo, Mich., dedicated to the memory of J. W. T. Youngs), Lecture Notes in Math., Springer, Berlin, 303, 111–124 (1972)
23.
Zurück zum Zitat H.F. Jensen, An upper bound for the rectilinear crossing number of the complete graph. J. Combin. Theor. Ser B 10, 212–216 (1971)MATHCrossRef H.F. Jensen, An upper bound for the rectilinear crossing number of the complete graph. J. Combin. Theor. Ser B 10, 212–216 (1971)MATHCrossRef
24.
Zurück zum Zitat H. Krasser, Order Types of Point Sets in the Plane. Ph.D. thesis, Institute for Theoretical Computer Science, Graz University of Technology, Austria, October 2003 H. Krasser, Order Types of Point Sets in the Plane. Ph.D. thesis, Institute for Theoretical Computer Science, Graz University of Technology, Austria, October 2003
25.
Zurück zum Zitat L. Lovász, K. Vesztergombi, U. Wagner, E. Welzl, Convex quadrilaterals and k-sets. Towards a theory of geometric graphs. Contemp. Math. (Amer. Math. Soc., Providence) 342, 139–148 (2004) L. Lovász, K. Vesztergombi, U. Wagner, E. Welzl, Convex quadrilaterals and k-sets. Towards a theory of geometric graphs. Contemp. Math. (Amer. Math. Soc., Providence) 342, 139–148 (2004)
27.
Zurück zum Zitat E. Scheinerman, H.S. Wilf, The rectilinear crossing number of a complete graph and Sylvester’s “four point” problem of geometric probability. Am. Math. Mon. 101, 939–943 (1994)MathSciNetMATHCrossRef E. Scheinerman, H.S. Wilf, The rectilinear crossing number of a complete graph and Sylvester’s “four point” problem of geometric probability. Am. Math. Mon. 101, 939–943 (1994)MathSciNetMATHCrossRef
28.
Zurück zum Zitat D. Singer, Rectilinear crossing numbers. Manuscript (1971) D. Singer, Rectilinear crossing numbers. Manuscript (1971)
29.
Zurück zum Zitat J.J. Sylvester, Question 1491, in The Educational Times (London, April 1864) J.J. Sylvester, Question 1491, in The Educational Times (London, April 1864)
30.
Zurück zum Zitat J.J. Sylvester, Rep. Br. Assoc. 35, 8–9 (1865) J.J. Sylvester, Rep. Br. Assoc. 35, 8–9 (1865)
31.
Zurück zum Zitat U. Wagner, On the rectilinear crossing number of complete graphs, in Proceedings of the 14th Annual Symposium on Discrete Algorithms (Baltimore, MD, 2003), ACM, New York, 583–588 (2003) U. Wagner, On the rectilinear crossing number of complete graphs, in Proceedings of the 14th Annual Symposium on Discrete Algorithms (Baltimore, MD, 2003), ACM, New York, 583–588 (2003)
Metadaten
Titel
The Rectilinear Crossing Number of K n : Closing in (or Are We?)
verfasst von
Bernardo M. Ábrego
Silvia Fernández-Merchant
Gelasio Salazar
Copyright-Jahr
2013
Verlag
Springer New York
DOI
https://doi.org/10.1007/978-1-4614-0110-0_2

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