

A014540


Rectilinear crossing number of complete graph on n nodes.


4



0, 0, 0, 0, 1, 3, 9, 19, 36, 62, 102, 153, 229, 324, 447, 603, 798, 1029, 1318, 1657, 2055, 2528, 3077, 3699, 4430, 5250, 6180
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OFFSET

1,6


COMMENTS

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


REFERENCES

M. Gardner, Crossing Numbers. Ch. 11 in Knotted Doughnuts and Other Mathematical Entertainments. New York: W. H. Freeman, 1986.
C. Thomassen, Embeddings and minors, pp. 301349 of R. L. Graham et al., eds., Handbook of Combinatorics, MIT Press.


LINKS

Table of n, a(n) for n=1..27.
B. M. Abrego, S. FernandezMerchant, J. LeaĆ±os and G. Salazar, The maximum number of halving lines and the rectilinear crossing number of K_n for n <= 27, Electronic Notes in Discrete Mathematics, 30 (2008), 261266.
O. Aichholzer, Crossing number project
O. Aichholzer, F. Aurenhammer and H. Krasser, Progress on rectilinear crossing numbers. [Broken link]
O. Aichholzer, F. Aurenhammer and H. Krasser, Progress on rectilinear crossing numbers, Technical report, IGITU Graz, Austria, 2001.
O. Aichholzer, F. Aurenhammer and H. Krasser, On the Rectilinear Crossing Number [Broken link]
O. Aichholzer, J. Garcia, D. Orden, P. Ramos, New lower bounds for the number of <= kedges and the rectilinear crossing number of K_n, Discrete & Computational Geometry 38 (2007), 114.
O. Aichholzer and H. Krasser, The point set order type data base: a collection of applications and results, pp. 1720 in Abstracts 13th Canadian Conference on Computational Geometry (CCCG '01), Waterloo, Aug. 1315, 2001. [Broken link]
D. Archdeacon, The rectilinear crossing number
D. Bienstock and N. Dean, Bounds for rectilinear crossing numbers, J. Graph Theory 17 (1993) 333348
A. Brodsky, S. Durocher and E. Gethner, The Rectilinear Crossing Number of K_{10} is 62, The Electronic J. Combin, #R23, 2001.
A. Brodsky, S. Durocher and E. Gethner, Toward the rectilinear crossing number of K_n: new drawings, upper bounds, and asymptotics, Discrete Math. 262 (2003), 5977.
D. Garber, The Orchard crossing number of an abstract graph, arXiv:math/0303317 [math.CO], 20032009.
H. F. Jensen, An Upper Bound for the Rectilinear Crossing Number of the Complete Graph, J. Comb. Th. Ser. B 10, 212216, 1971.
Eric Weisstein's World of Mathematics, Graph Crossing Number
Eric Weisstein's World of Mathematics, Rectilinear Crossing Number.
Eric Weisstein's World of Mathematics, Zarankiewicz's Conjecture


CROSSREFS

Cf. A000241, A030179, A006247.
Sequence in context: A062748 A147174 A147158 * A146694 A146050 A147500
Adjacent sequences: A014537 A014538 A014539 * A014541 A014542 A014543


KEYWORD

nonn,nice,hard,more,changed


AUTHOR

Eric W. Weisstein


EXTENSIONS

102 from Oswin Aichholzer (oswin.aichholzer(AT)tugraz.at), Aug 14 2001
153 from Hannes Krasser (hkrasser(AT)igi.tugraz.ac.at), Sep 17 2001
More terms from Eric W. Weisstein, Nov 30 2006
More terms from Bernardo M. Abrego (bernardo.abrego(AT)csun.edu), May 05 2008


STATUS

approved



