Abstract
Let 
(G) denote the rectilinear crossing number of a graph G. We determine (K11)=102 and (K12)=153. Despite the remarkable hunt for crossing numbers of the complete graph K
n
– initiated by R. Guy in the 1960s – these quantities have been unknown forn>10 to date. Our solution mainly relies on a tailor-made method for enumerating all inequivalent sets of points (order types) of size 11.
Based on these findings, we establish a new upper bound on (K
n
) for general n. The bound stems from a novel construction of drawings of K
n
with few crossings.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ábrego, B., Fernández-Merchant, S.: A lower bound for the rectilinear crossing number. Graphs and Combinatorics (to appear).
O. Aichholzer F. Aurenhammer H. Krasser (2002) ArticleTitleEnumerating order types for small point sets with applications Order 19 265–281 Occurrence Handle10.1023/A:1021231927255
Aichholzer, O., Aurenhammer, F., Krasser, H.: On the crossing number of complete graphs. Proc. 18th Ann. ACM Symp. Comput. Geometry, Barcelona, Spain, 2002, pp. 19–24.
Aichholzer, O., Krasser, H.: The point set order type data base: a collection of applications and results. Proc. 13th Ann. Canadian Conf. Comput. Geometry CCCG 2001, Waterloo, Canada, 2001, pp. 17–20.
Aichholzer, O., Krasser, H.: Abstract order type extension and new results on the rectilinear crossing number. Proc. 21th Ann. ACM Symp. Computational Geometry 2005, Pise, Italy, pp. 91–98.
Balogh, J., Salazar, G.: On k-sets, convex quadrilaterals, and the rectilinear crossing number of K n (submitted).
D. Bienstock (1991) ArticleTitleSome provably hard crossing number problems Discrete Comput. Geometry 6 443–459
Brodsky, A., Durocher, S., Gether, E.: The rectilinear crossing number of K10 is 62. The Electronic J. Combinatorics 8, Research Paper 23 (2001).
A. Brodsky S. Durocher E. Gether (2003) ArticleTitleToward the rectilinear crossing number of K n new drawings, upper bounds, and asymptotics Discrete Math. 262 59–77 Occurrence Handle10.1016/S0012-365X(02)00491-0
P. Erdös R. K. Guy (1973) ArticleTitleCrossing number problems Am. Math. Month. 88 52–58
R. K. Garey D. S. Johnson (1983) ArticleTitleCrossing number is NP-complete SIAM J. Alg. Disc. Meth. 4 312–316
J. E. Goodman R. Pollack (1983) ArticleTitleMulti-dimensional sorting SIAM J. Comput. 12 484–507 Occurrence Handle10.1137/0212032
R. K. Guy (1960) ArticleTitleA combinatorial problem Nabla (Bull. Malayan Math. Soc.) 7 68–72
F. Harary A. Hill (1962) ArticleTitleOn the number of crossings in a complete graph Proc. Edinburgh Math. Soc. 13 IssueID2 333–338
R. B. Hayward (1987) ArticleTitleA lower bound for the optimal crossing-free Hamiltonian cycle problem Discrete Comput. Geometry 2 327–343
H. F. Jensen (1971) ArticleTitleAn upper bound for the rectilinear crossing number of the complete graph J. Combinatorial Theory B 29 212–216 Occurrence Handle10.1016/0095-8956(71)90045-1
F. T. Leighton (1983) Complexity issues in VLSI The MIT Press Cambridge
Lovász, L., Vesztergombi, K., Welzl, E., Wagner, U.: Convex quadrilaterals and k-sets. In: Towards a theory of geometric graphs (Pach J. ed.). AMS Contemporary Mathematics Ser. 342, 139–148 (2004).
J. Pach G. Tóth (2000) ArticleTitleWhich crossing number is it, anyway? J. Combinatorial Theory B 80 225–246 Occurrence Handle10.1006/jctb.2000.1978
J. Pach G. Tóth (2000) ArticleTitleThirteen problems on crossing numbers Geombinatorics 9 195–207
R. B. Richter C. Thomassen (1997) ArticleTitleRelations between crossing numbers of complete and complete bipartite graphs Am. Math. Month. 104 131–137
E. R. Scheinerman H. S. Wilf (1994) ArticleTitleThe rectilinear crossing number of a complete graph and Sylvester's “four point problem” of geometric probability Am. Math. Month. 101 939–943
Singer, D.: The rectilinear crossing number of certain graphs. Manuscript, 1971. Available at: http://www.cwru.edu/artsci/math/singer/.
J. T. Thorpe F. C. Harris (1996) ArticleTitleA parallel stochastic optimization algorithm for finding mappings of the rectilinear minimal crossing number Ars Combinatorica 43 135–148
W. T. Tutte (1970) ArticleTitleToward a theory of crossing numbers J. Combinatorial Theory 8 45–53
Wagner, U.: On the rectilinear crossing number of complete craphs. Proc. 14th Ann. Symp. on Discrete Algorithms (SODA), 2003, pp. 583–588.
Wilf, H. S.: On crossing numbers, and some unsolved problems. In: Combinatorics, geometry, and probability. Cambridge: Cambridge University Press 1997, pp. 557–562
http://www.ist.tugraz.at/aichholzer/crossings.html.
http://www.igi.tugraz.at/hkrasser/data/d36.nb.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Aichholzer, O., Aurenhammer, F. & Krasser, H. On the Crossing Number of Complete Graphs. Computing 76, 165–176 (2006). https://doi.org/10.1007/s00607-005-0133-3
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00607-005-0133-3