点分裂圆的数目

大圆排列图的三种可染性

Three Colorability of an Arrangement Graph of Great Circles

2000年，Stan Wagon提出了以下问题。作为平面地图来看，每个带面体的面都是3-着色的吗？一个等价的问题，在不同的伪装下，是下面的：是安排图的大圆圈的领域总是顶点3-着色？(The布置图具有用于每个交点的顶点和用于直接连接两个交点的每个弧的边。假设没有三个圆相交于一点，所以这个排列图是4-正则的。在本文中，我们证明了所有如上定义的排列图都是3-可着色的。

平面点集确定的圆

Circles determined by planar point sets

该研究确定了平面上非共线非共圆n点集确定的含三点不同圆的最小数量c(n)，除三个特殊阶数外等于F(n)，还解决了无三点共线的变体问题，相关工作由人类与AI合作完成。

Small 4-regular planar graphs that are not circle representable

A 4-regular planar graph $G$ is said to be circle representable if there exists a collection of circles drawn on the plane such that the touching and crossing points correspond to the vertices of $G$, and the circular arcs between those points correspond to the edges of $G$. Lovász (1970) conjectured that every 4-regular planar graph has a circle representation, but an infinite family of counterexamples was given by Bekos and Raftopoulou (2015). We reduce the order of the smallest known counterexamples among simple graphs from 822 to 68 based on a multigraph counterexample of order 12.

Stability of Erdős-Ko-Rado Theorems in Circle Geometries

Circle geometries are incidence structures that capture the geometry of circles on spheres, cones and hyperboloids in 3-dimensional space. In a previous paper, the author characterised the largest intersecting families in finite ovoidal circle geometries, except for Möbius planes of odd order. In this paper we show that also in these Möbius planes, if the order is greater than 3, the largest intersecting families are the sets of circles through a fixed point. We show the same result in the only known family of finite non-ovoidal circle geometries. Using the same techniques, we show a stability…

减半圈的数量

The number of halving circles

平面上2n+1个点的集合S称为在一般位置，如果S中没有三个点共线，也没有四个点共圈。一个圆被称为关于S的二等分，如果它的圆周上有三个S的点，内部有n-1个点，外部有n-1个点。我们证明了以下令人惊讶的结果：平面上任意2n+1个点的集合都恰好有n^2个等分圆。

On the number of intersection points of lines and circles in $\mathbb R^3$

We consider the following question: Given $n$ lines and $n$ circles in $\mathbb{R}^3$, what is the maximum number of intersection points lying on at least one line and on at least one circle of these families. We prove that if there are no $n^{1/2}$ curves (lines or circles) lying on an algebraic surface of degree at most two, then the number of these intersection points is $O(n^{3/2})$.

圆上具有两个锚点的标记点构型空间

Configuration spaces of labeled points on a circle with two anchors

本文研究圆上含两个固定锚点的 $n$ 个标记点构型空间，利用离散 Morse 理论确定其 Betti 数，并给出同调与上同调基的显式组合描述。

用圆分离点

Separating dots with circles

本文研究在平面或球面上用两类圆分离一般位置有限点集的方法数，证明了分离点集的圆的数量与点构型无关并给出显式公式，同时揭示了点构型连续变化时高阶 Voronoi 分解的局部移动规律及其对 cluster algebra 的影响。

$n$-arc and $n$-circle connected graph-like spaces

A space $X$ is $n$-arc connected (respectively, $n$-circle connected) if for any choice of at most $n$ points there is an arc (respectively, a circle) in $X$ containing the specified points. We study $n$-arc connectedness and $n$-circle connectedness in compactifications of locally finite graphs and the slightly more general class of graph-like continua, uncovering a striking difference in their behaviour regarding $n$-arc and -circle connectedness.

均匀映射曲面上的等距点圆构形

Isometric point-circle configurations on surfaces from uniform maps

我们将图的邻域几何嵌入曲面上作为点圆构型。我们给出的例子来自正规映射的表面上的最大数量的自同构为他们的属和调查的几何实现的五边形几何来自摩尔图。给出了具有两个$p$ -仿射的$p$ -仿射曲面上的点圆$v_4$构型的无限族.本文还描述了这些组态在两个$p$ -态射下在球面上的象.

On sets defining few ordinary circles

An ordinary circle of a set $P$ of $n$ points in the plane is defined as a circle that contains exactly three points of $P$. We show that if $P$ is not contained in a line or a circle, then $P$ spans at least $\frac{1}{4}n^2 - O(n)$ ordinary circles. Moreover, we determine the exact minimum number of ordinary circles for all sufficiently large $n$ and describe all point sets that come close to this minimum. We also consider the circle variant of the orchard problem. We prove that $P$ spans at most $\frac{1}{24}n^3 - O(n^2)$ circles passing through exactly four points of $P$. Here we determine …

杯积与具有两个锚点的圆的构型空间的更高拓扑复杂度

Cup products and the higher topological complexity of configuration spaces of the circle with two anchored points

研究利用离散 Morse 理论计算双锚点圆的锚定构型空间杯积，并据此给出在 $s$ 足够大时精确的更高拓扑复杂度 $TC_s$ 界。

Distinct Distances Between a Circle and a Generic Set

Let $S$ be a set of points in $\mathbb{R}^2$ contained in a circle and $P$ an unrestricted point set in $\mathbb{R}^2$. We prove the number of distinct distances between points in $S$ and points in $P$ is at least $\min(|S||P|^{1/4-\varepsilon},|S|^{2/3}|P|^{2/3},|S|^2,|P|^2)$. This builds on work of Pach and De Zeeuw, Bruner and Sharir, McLaughlin and Omar and Mathialagan on distances between pairs of sets.

关于普通圆的个数

On the number of ordinary circles

我们证明，任何$n$点在$\mathbb{R}^2$，而不是所有的线或圆，确定至少$\frac{1}{4}n^2-O(n)$普通的圆（圆恰好包含三个的$n$点）。这个界的主项对于甚至$n$是最可能的。我们的证明依赖于最近的结果绿色和陶普通线。

Delaunay Triangulations of Points on Circles

Delaunay triangulations of a point set in the Euclidean plane are ubiquitous in a number of computational sciences, including computational geometry. Delaunay triangulations are not well defined as soon as 4 or more points are concyclic but since it is not a generic situation, this difficulty is usually handled by using a (symbolic or explicit) perturbation. As an alternative, we propose to define a canonical triangulation for a set of concyclic points by using a max-min angle characterization of Delaunay triangulations. This point of view leads to a well defined and unique triangulation as lo…

Miquel dynamics for circle patterns

We study a new discrete-time dynamical system on circle patterns with the combinatorics of the square grid. This dynamics, called Miquel dynamics, relies on Miquel's six circles theorem. We provide a coordinatization of the appropriate space of circle patterns on which the dynamics acts and use it to derive local recurrence formulas. Isoradial circle patterns arise as periodic points of Miquel dynamics. Furthermore, we prove that certain signed sums of intersection angles are preserved by the dynamics. Finally, when the initial circle pattern is spatially biperiodic with a fundamental domain o…