On the 2-colored crossing number

关于单位正方形中随机点集的2-着色直线交叉数的注记

A Note on the 2-Colored Rectilinear Crossing Number of Random Point Sets in the Unit Square

本文研究单位正方形中随机四点集的边着色交叉问题，证明同色交叉边概率为1/4，并给出随机绘制下使单色交叉数期望降至交叉总数1/2-7/50的着色方案。

关于稠密几何图的k色交叉比的一个注记

A Note on the $k$-colored Crossing Ratio of Dense Geometric Graphs

本文证明每个稠密几何图的边可用k种颜色着色，使同色相交边对比例至多为1/k减去某正常数，推广了Aichholzer等人关于完全几何图k=2的结果。

关于 $K_n$ 的几何 $k$-着色交叉数

On the geometric $k$-colored crossing number of $K_n$

本文研究完全图 $K_n$ 的几何 $k$-着色交叉数，提出一种通用程序，从低交叉初始绘图构造任意大 $n$ 的 $k$-边着色绘图，大幅改进 $k=2$ 至 $10$ 的渐近上界，并借助 MAX-$k$-CUT 公式进行启发式搜索。

Colored Non-Crossing Euclidean Steiner Forest

Given a set of $k$-colored points in the plane, we consider the problem of finding $k$ trees such that each tree connects all points of one color class, no two trees cross, and the total edge length of the trees is minimized. For $k=1$, this is the well-known Euclidean Steiner tree problem. For general $k$, a $kρ$-approximation algorithm is known, where $ρ\le 1.21$ is the Steiner ratio. We present a PTAS for $k=2$, a $(5/3+\varepsilon)$-approximation algorithm for $k=3$, and two approximation algorithms for general~$k$, with ratios $O(\sqrt n \log k)$ and $k+\varepsilon$.

Colored ray configurations

We study the cyclic color sequences induced at infinity by colored rays with apices being a given balanced finite bichromatic point set. We first study the case in which the rays are required to be pairwise disjoint. We derive a lower bound on the number of color sequences that can be realized from any such fixed point set and examine color sequences that can be realized regardless of the point set, exhibiting negative examples as well. We also provide a tight upper bound on the number of configurations that can be realized from a point set, and point sets for which there are asymptotically le…

Long Alternating Paths Exist

Let $P$ be a set of $2n$ points in convex position, such that $n$ points are colored red and $n$ points are colored blue. A non-crossing alternating path on $P$ of length $\ell$ is a sequence $p_1, \dots, p_\ell$ of $\ell$ points from $P$ so that (i) all points are pairwise distinct; (ii) any two consecutive points $p_i$, $p_{i+1}$ have different colors; and (iii) any two segments $p_i p_{i+1}$ and $p_j p_{j+1}$ have disjoint relative interiors, for $i \neq j$. We show that there is an absolute constant $\varepsilon > 0$, independent of $n$ and of the coloring, such that $P$ always admits a no…