论文标题
完整图和拓扑性heilbronn问题的简单图纸中的脱节面
Disjoint faces in simple drawings of the complete graph and topological Heilbronn problems
论文作者
论文摘要
鉴于一个完整的简单拓扑图$ g $，$ g $生成的$ k $ face是由$ g $的非自动相互交流$ k $ cycle的边缘所包围的开放式区域。有趣的是，它具有完整的简单拓扑图，其生成的每个奇数都包含原点。在本文中，我们表明，每个完整的$ n $ vertex简单拓扑图都至少生成$ω（n^{1/3}）$成对脱节4 face。作为立即推论，在单位广场上绘制的$ N $顶点上的每个完整的简单拓扑图都会产生一个4脸，最多最多有$ o（n^{ - 1/3}）$。最后，我们研究了Heilbronn三角问题的$ \ Mathbb Z_2 $变体。
Given a complete simple topological graph $G$, a $k$-face generated by $G$ is the open bounded region enclosed by the edges of a non-self-intersecting $k$-cycle in $G$. Interestingly, there are complete simple topological graphs with the property that every odd face it generates contains the origin. In this paper, we show that every complete $n$-vertex simple topological graph generates at least $Ω(n^{1/3})$ pairwise disjoint 4-faces. As an immediate corollary, every complete simple topological graph on $n$ vertices drawn in the unit square generates a 4-face with area at most $O(n^{-1/3})$. Finally, we investigate a $\mathbb Z_2$ variant of Heilbronn triangle problem.
