The table below shows the best known constructions for the rectilinear crossing numbers of complete graphs of order n, for small values of n. The values for n up to 12 are known to be optimal. The values up to n = 9 have been known for some time. I completed a computational proof for 10, 11 and 12 several years ago. More recently, an analytical proof for n=10 appeared in EJC.
For more recent, and better, results click here.
| n | Crossings | Link to plot file |
| 10 | 62 | Points |
| 11 | 102 | Points |
| 12 | 153 | Points |
| 13 | 229 | Points |
| 14 | 324 | Points |
| 15 | 447 | Points |
| 16 | 603 | Points |
| 17 | 798 | Points |
| 18 | 1029 | Points |
| 19 | 1318 | Points |
| 20 | 1657 | Points |
| 21 | 2055 | Points |
| 22 | 2529 | Points |
| 23 | 3079 | Points |
| 24 | 3702 | Points |
| 25 | 4432 | Points |
| 30 | 9734 | Points |
Geoff Exoo