关于$K_n$的3-可分解几何图
On 3-decomposable geometric drawings of $K_n$
摘要：
所有已知的$K_n$的最佳直线图的点集共享一个明确无误的聚类属性，即所谓的{ \em 3-可分解性}。人们普遍认为$K_n$的所有最优直线图的基本点集都是3-可分解的。本文给出了3-可分解$n$ --点集的最小$(\le k)$--集个数的一个下界.作为直接推论，我们得到了$K_n$的任意直线图$\dd$与其基础3-可分解点集的交叉数$\rcr(\dd)$的一个下界，即$\rcr(\dd) > {2/27}(15-π^{2})\binom{n}{4}+Θ(n^{3}) \approx 0.380029\binom{n}{4} + Θ(n^3)$ .在3-可分解性的假设下，这将$K_n$的直线交叉数$\rcr(K_n)$的最佳已知下限和上限之间的差距缩小了40%以上。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
