On the Crossing Number of Complete Graphs

Authors:
O. Aichholzer;F. Aurenhammer;H. Krasser
Affiliations:
Institute for Software Technology, Graz University of Technology, Inffeldgasse 16b, 8010, Graz, Austria;Institute for Theoretical Computer Science, Graz University of Technology, Inffeldgasse 16b, 8010, Graz, Austria;Institute for Theoretical Computer Science, Graz University of Technology, Inffeldgasse 16b, 8010, Graz, Austria
Venue:
Computing
Year:
2006

Citing 0
Cited 4

Note: Geometric drawings of Kn with few crossings

Journal of Combinatorial Theory Series A
Note: An extended lower bound on the number of (≤k)-edges to generalized configurations of points and the pseudolinear crossing number of Kn

Journal of Combinatorial Theory Series A
The complexity of several realizability problems for abstract topological graphs

GD'07 Proceedings of the 15th international conference on Graph drawing
3-symmetric and 3-decomposable geometric drawings of Kn

Discrete Applied Mathematics

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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 Kn – initiated by R. Guy in the 1960s – these quantities have been unknown forn10 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 **(Kn) for general n. The bound stems from a novel construction of drawings of Kn with few crossings.