Academic Journal
On the Crossing Number of Complete Graphs.
| Title: | On the Crossing Number of Complete Graphs. |
|---|---|
| Authors: | Aichholzer, O.1 oaich@ist.tugraz.at, Aurenhammer, F.2 auren@igi.tugraz.at, Krasser, H.2 hkrasser@igi.tugraz.at |
| Source: | Computing. Jan2006, Vol. 76 Issue 1/2, p165-176. 12p. 2 Diagrams, 3 Charts. |
| Abstract: | Let [InlineMediaObject not available: see fulltext.]( G) denote the rectilinear crossing number of a graph G. We determine [InlineMediaObject not available: see fulltext.]( K 11)=102 and [InlineMediaObject not available: see fulltext.]( K 12)=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 for n>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 [InlineMediaObject not available: see fulltext.]( K n ) for general n. The bound stems from a novel construction of drawings of K n with few crossings. [ABSTRACT FROM AUTHOR] |
| Subject Terms: | *Graphic methods, *Linear statistical models, Graph theory, Set theory, Linear algebra |
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| ISSN: | 0010485X |
| DOI: | 10.1007/s00607-005-0133-3 |
| Database: | Business Source Complete |