On 3-decomposable geometric drawings of $K_n$
| dc.creator | Abrego, Bernardo | |
| dc.creator | Fernandez-Merchant, Silvia | |
| dc.creator | Leanos, Jesus | |
| dc.creator | Salazar, Gelasio | |
| dc.date | 2007-12-27 | |
| dc.date.accessioned | 2026-07-07T08:51:25Z | |
| dc.date.available | 2026-07-07T08:51:25Z | |
| dc.description | The point sets of all known optimal rectilinear drawings of $K_n$ share an unmistakeable clustering property, the so--called {\em 3--decomposability}. It is widely believed that the underlying point sets of all optimal rectilinear drawings of $K_n$ are 3--decomposable. We give a lower bound for the minimum number of $(\le k)$--sets in a 3--decomposable $n$--point set. As an immediate corollary, we obtain a lower bound for the crossing number $\rcr(\dd)$ of any rectilinear drawing $\dd$ of $K_n$ with underlying 3--decomposable point set, namely $\rcr(\dd) > {2/27}(15-π^{2})\binom{n}{4}+Θ(n^{3}) \approx 0.380029\binom{n}{4} + Θ(n^3)$. This closes this gap between the best known lower and upper bounds for the rectilinear crossing number $\rcr(K_n)$ of $K_n$ by over 40%, under the assumption of 3--decomposability. | |
| dc.identifier | https://arxiv.org/abs/0712.4255 | |
| dc.identifier | http://arxiv.org/abs/0712.4255 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/144930 | |
| dc.subject | Combinatorics | |
| dc.subject | 05C10 | |
| dc.title | On 3-decomposable geometric drawings of $K_n$ | |
| dc.type | text |