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%I #33 Jan 22 2018 15:49:01
%S 0,0,0,0,1,3,9,19,36,62,102,153,229,324,447,603,798,1029,1318,1657,
%T 2055,2528,3077,3699,4430,5250,6180
%N Rectilinear crossing number of complete graph on n nodes.
%C The values a(19) and a(21) were obtained by Aichholzer et al. in 2006. The value a(18) is claimed by the Rectilinear Crossing Number project after months of distributed computing. This was confirmed by Abrego et al., they also found the values a(20) and a(22) to a(27). The next unknown entry, a(28), is either 7233 or 7234. - Bernardo M. Abrego (bernardo.abrego(AT)csun.edu), May 05 2008
%D M. Gardner, Crossing Numbers. Ch. 11 in Knotted Doughnuts and Other Mathematical Entertainments. New York: W. H. Freeman, 1986.
%D C. Thomassen, Embeddings and minors, pp. 301-349 of R. L. Graham et al., eds., Handbook of Combinatorics, MIT Press.
%H B. M. Abrego, S. Fernandez-Merchant, J. Leaños and G. Salazar, <a href="http://dx.doi.org/10.1016/j.endm.2008.01.045">The maximum number of halving lines and the rectilinear crossing number of K_n for n <= 27</a>, Electronic Notes in Discrete Mathematics, 30 (2008), 261-266.
%H O. Aichholzer, <a href="http://www.ist.tugraz.at/staff/aichholzer/crossings.html">Crossing number project</a>
%H O. Aichholzer, F. Aurenhammer and H. Krasser, <a href="http://www.igi.TUGraz.at/oaich/psfiles/aak-prcn-01.ps.gz">Progress on rectilinear crossing numbers</a>. [Broken link]
%H O. Aichholzer, F. Aurenhammer and H. Krasser, <a href="http://www.igi.tugraz.at/Abstracts/aak-prcn-01/">Progress on rectilinear crossing numbers</a>, Technical report, IGI-TU Graz, Austria, 2001.
%H O. Aichholzer, F. Aurenhammer and H. Krasser, <a href="http://www.igi.TUGraz.at/oaich/triangulations/crossing.html">On the Rectilinear Crossing Number</a> [Broken link]
%H O. Aichholzer, J. Garcia, D. Orden, P. Ramos, <a href="http://dx.doi.org/10.1007/s00454-007-1325-8">New lower bounds for the number of <= k-edges and the rectilinear crossing number of K_n</a>, Discrete & Computational Geometry 38 (2007), 1-14.
%H O. Aichholzer and H. Krasser, <a href="http://www.ist.tugraz.at/publications/oaich/psfiles/ak-psotd-01.ps.gz">The point set order type data base: a collection of applications and results</a>, pp. 17-20 in Abstracts 13th Canadian Conference on Computational Geometry (CCCG '01), Waterloo, Aug. 13-15, 2001. [Broken link]
%H D. Archdeacon, <a href="http://www.emba.uvm.edu/~archdeac/problems/rectcros.htm">The rectilinear crossing number</a>
%H D. Bienstock and N. Dean, <a href="http://dx.doi.org/10.1002/jgt.3190170308">Bounds for rectilinear crossing numbers</a>, J. Graph Theory 17 (1993) 333-348
%H A. Brodsky, S. Durocher and E. Gethner, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v8i1r23">The Rectilinear Crossing Number of K_{10} is 62</a>, The Electronic J. Combin, #R23, 2001.
%H A. Brodsky, S. Durocher and E. Gethner, <a href="http://dx.doi.org/10.1016/S0012-365X(02)00491-0">Toward the rectilinear crossing number of K_n: new drawings, upper bounds, and asymptotics</a>, Discrete Math. 262 (2003), 59-77.
%H D. Garber, <a href="http://arXiv.org/abs/math.CO/0303317">The Orchard crossing number of an abstract graph</a>, arXiv:math/0303317 [math.CO], 2003-2009.
%H H. F. Jensen, <a href="http://dx.doi.org/10.1016/0095-8956(71)90045-1">An Upper Bound for the Rectilinear Crossing Number of the Complete Graph</a>, J. Comb. Th. Ser. B 10, 212-216, 1971.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GraphCrossingNumber.html">Graph Crossing Number</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RectilinearCrossingNumber.html">Rectilinear Crossing Number</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ZarankiewiczsConjecture.html">Zarankiewicz's Conjecture</a>
%Y Cf. A000241, A030179, A006247.
%K nonn,nice,hard,more
%O 1,6
%A _Eric W. Weisstein_
%E 102 from Oswin Aichholzer (oswin.aichholzer(AT)tugraz.at), Aug 14 2001
%E 153 from Hannes Krasser (hkrasser(AT)igi.tu-graz.ac.at), Sep 17 2001
%E More terms from _Eric W. Weisstein_, Nov 30 2006
%E More terms from Bernardo M. Abrego (bernardo.abrego(AT)csun.edu), May 05 2008
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