论文标题
伪圆的布置：在Digon和Triangles上
Arrangements of Pseudocircles: On Digons and Triangles
论文作者
论文摘要
在本文中，我们研究了成对相交伪圆的简单排列的细胞结构。重点将放在格伦鲍姆（Grünbaum）（1972）的两个问题上。 首先，我们讨论最大数量的Digon或接触点。格伦鲍姆（Grünbaum）猜想，最多有$ 2n-2 $ digon牢房，或者最多只有$ 2N-2 $触摸。 Agarwal等。 （2004年）验证了圆柱形布置的猜想。我们表明，猜想的任何布置都包含三个成对触摸伪圆的任何布置。证明将结果用于圆柱形布置。此外，我们构建了非圆柱形布置，该布置的最高为$ 2N-2 $触摸，并且没有三倍的成对触摸伪圆。 其次，我们讨论了没有digon和触摸的排列中三角细胞（三角形）的最小数量。格伦鲍姆（Grünbaum）猜想，这种安排的三角形为2n-4美元。 Snoeyink和Hershberger（1991）建立了$ \ lceil \ frac {4} {3} n \ rceil $的下限。 Felsner和Scheucher（2017）反驳了猜想，并仅使用$ \ lceil \ frac {16} {11} {11} n \ rceil $ triangles构建了一个安排。我们提供了一个结构，该结构表明$ \ lceil \ frac {4} {3} n \ rceil $是正确的值。
In this article, we study the cell-structure of simple arrangements of pairwise intersecting pseudocircles. The focus will be on two problems of Grünbaum (1972). First, we discuss the maximum number of digons or touching points. Grünbaum conjectured that there are at most $2n - 2$ digon cells or equivalently at most $2n - 2$ touchings. Agarwal et al. (2004) verified the conjecture for cylindrical arrangements. We show that the conjecture holds for any arrangement which contains three pairwise touching pseudocircles. The proof makes use of the result for cylindrical arrangements. Moreover, we construct non-cylindrical arrangements which attain the maximum of $2n - 2$ touchings and have no triple of pairwise touching pseudocircles. Second, we discuss the minimum number of triangular cells (triangles) in arrangements without digons and touchings. Grünbaum conjectured that such arrangements have $2n - 4$ triangles. Snoeyink and Hershberger (1991) established a lower bound of $\lceil \frac{4}{3}n \rceil$. Felsner and Scheucher (2017) disproved the conjecture and constructed a family of arrangements with only $\lceil \frac{16}{11}n \rceil$ triangles. We provide a construction which shows that $\lceil \frac{4}{3}n \rceil$ is the correct value.
