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A014540 Rectilinear crossing number of complete graph on n nodes. 3
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 (list; graph; refs; listen; history; text; internal format)
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

O. Aichholzer, F. Aurenhammer and H. Krasser. Progress on rectilinear crossing numbers. Technical report, IGI-TU Graz, Austria, 2001.

D. Bienstock and N. Dean, Bounds for rectilinear crossing numbers, J. Graph Theory 17 (1993) 333-348

A. Brodsky, S. Durocher and E. Gethner, Toward the rectilinear crossing number of K_n..., Discrete Math. 262 (2003), 59-77.

M. Gardner, Crossing Numbers. Ch. 11 in Knotted Doughnuts and Other Mathematical Entertainments. New York: W. H. Freeman, 1986.

H. F. Jensen, An Upper Bound for the Rectilinear Crossing Number of the Complete Graph, J. Comb. Th. Ser. B 10, 212-216, 1971.

C. Thomassen, Embeddings and minors, pp. 301-349 of R. L. Graham et al., eds., Handbook of Combinatorics, MIT Press.

B.M. Abrego, S. Fernandez-Merchant, J. Leanos 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), 261-266.

O. Aichholzer, J. Garcia, D. Orden, P. Ramos, New lower bounds for the number of <= k-edges and the rectilinear crossing number of K_n. Discrete & Computational Geometry 38 (2007), 1-14.

LINKS

Table of n, a(n) for n=1..27.

O. Aichholzer, Crossing number project

O. Aichholzer, F. Aurenhammer and H. Krasser, Progress on rectilinear crossing numbers.

O. Aichholzer, F. Aurenhammer and H. Krasser, On the Rectilinear Crossing Number

O. Aichholzer and H. Krasser, The point set order type data base: a collection of applications and results, pp. 17-20 in Abstracts 13-th Canadian Conference on Computational Geometry (CCCG '01), Waterloo, Aug. 13-15, 2001.

D. Archdeacon, The rectilinear crossing number

A. Brodsky, S. Durocher and E. Gethner, The Rectilinear Crossing Number of K_{10} is 62, Electronic J. Combin., #R23, 2001.

D. Garber, The Orchard crossing number of an abstract graph

Eric Weisstein's World of Mathematics, Graph Crossing Number

Eric Weisstein's World of Mathematics, Rectilinear Crossing Number.

E. Weisstein, Zarankiewicz's Conjecture

CROSSREFS

Cf. A000241, A030179.

Sequence in context: A062748 A147174 A147158 * A146694 A146050 A147500

Adjacent sequences:  A014537 A014538 A014539 * A014541 A014542 A014543

KEYWORD

nonn,nice,hard,more

AUTHOR

Eric W. Weisstein

EXTENSIONS

102 from Oswin Aichholzer (oswin.aichholzer(AT)tugraz.at), Aug 14, 2001

153 from Hannes Krasser (hkrasser(AT)igi.tu-graz.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

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Last modified April 19 05:22 EDT 2014. Contains 240738 sequences.