On the Crossing Numbers of Complete Graphs
In this thesis we prove two main results. The Triangle Conjecture asserts that the convex hull of any optimal rectilinear drawing of <em>K<sub>n</sub></em> must be a triangle (for <em>n</em> ≥ 3). We prove that, for the larger class of pseudolinear dr...
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University of Waterloo
2006
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ndltd-WATERLOO-oai-uwspace.uwaterloo.ca-10012-11742013-01-08T18:49:25ZPan, Shengjun2006-08-22T14:26:51Z2006-08-22T14:26:51Z20062006http://hdl.handle.net/10012/1174In this thesis we prove two main results. The Triangle Conjecture asserts that the convex hull of any optimal rectilinear drawing of <em>K<sub>n</sub></em> must be a triangle (for <em>n</em> ≥ 3). We prove that, for the larger class of pseudolinear drawings, the outer face must be a triangle. The other main result is the next step toward Guy's Conjecture that the crossing number of <em>K<sub>n</sub></em> is $(1/4)[n/2][(n-1)/2][(n-2)/2][(n-3)/2]$. We show that the conjecture is true for <em>n</em> = 11,12; previously the conjecture was known to be true for <em>n</em> ≤ 10. We also prove several minor results.application/pdf414477 bytesapplication/pdfenUniversity of WaterlooCopyright: 2006, Pan, Shengjun. All rights reserved.Mathematicsgraphcrossing numberGuy's ConjectureOn the Crossing Numbers of Complete GraphsThesis or DissertationCombinatorics and OptimizationMaster of Mathematics |
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NDLTD |
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en |
| format |
Others
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NDLTD |
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Mathematics graph crossing number Guy's Conjecture |
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Mathematics graph crossing number Guy's Conjecture Pan, Shengjun On the Crossing Numbers of Complete Graphs |
| description |
In this thesis we prove two main results. The Triangle Conjecture asserts that the convex hull of any optimal rectilinear drawing of <em>K<sub>n</sub></em> must be a triangle (for <em>n</em> ≥ 3). We prove that, for the larger class of pseudolinear drawings, the outer face must be a triangle. The other main result is the next step toward Guy's Conjecture that the crossing number of <em>K<sub>n</sub></em> is $(1/4)[n/2][(n-1)/2][(n-2)/2][(n-3)/2]$. We show that the conjecture is true for <em>n</em> = 11,12; previously the conjecture was known to be true for <em>n</em> ≤ 10. We also prove several minor results. |
| author |
Pan, Shengjun |
| author_facet |
Pan, Shengjun |
| author_sort |
Pan, Shengjun |
| title |
On the Crossing Numbers of Complete Graphs |
| title_short |
On the Crossing Numbers of Complete Graphs |
| title_full |
On the Crossing Numbers of Complete Graphs |
| title_fullStr |
On the Crossing Numbers of Complete Graphs |
| title_full_unstemmed |
On the Crossing Numbers of Complete Graphs |
| title_sort |
on the crossing numbers of complete graphs |
| publisher |
University of Waterloo |
| publishDate |
2006 |
| url |
http://hdl.handle.net/10012/1174 |
| work_keys_str_mv |
AT panshengjun onthecrossingnumbersofcompletegraphs |
| _version_ |
1716572472035246080 |