Edge-Removal and Non-Crossing Configurations in Geometric Graphs

Suppose that nk points in general position in the plane are colored red and blue, with at least n points of each color. We show that then there exist n pairwise disjoint convex sets, each of them containing k of the points, and each of them containing points of both colors.We also show that if P is a set of n(d + 1) points in general position in R d colored by d colors with at least n points of each color, then there exist n pairwise disjoint d-dimensional simplices with vertices in P , each of them containing a point of every color.These results can be viewed as a step towards a common generalization of several previously known geometric partitioning results regarding colored point sets.

Section: Introductionmentioning

confidence: 58%

Section: Merging Colorsmentioning

confidence: 99%

Near equipartitions of colored point sets

Holmsen

Kynčl

Valculescu

2017

“…Aichholzer et al [1] proved Theorem 2 by the same metric argument as the in case of two colors. Kano, Suzuki and Uno [13] first proved Theorem 2 for three colors, by induction using a result on partitions of 3-colored point sets on a line.…”

Section: Introductionmentioning

confidence: 80%

“…Aichholzer et al [1] and Kano, Suzuki and Uno [13] extended the planar version of Theorem 1 to an arbitrary number of colors as follows.…”

Section: Introductionmentioning

confidence: 99%

The hamburger theorem

Kanō

Kynčl

2018

We generalize the ham sandwich theorem to $d+1$ measures in $\mathbb{R}^d$ as follows. Let $\mu_1,\mu_2, \dots, \mu_{d+1}$ be absolutely continuous finite Borel measures on $\mathbb{R}^d$. Let $\omega_i=\mu_i(\mathbb{R}^d)$ for $i\in [d+1]$, $\omega=\min\{\omega_i; i\in [d+1]\}$ and assume that $\sum_{j=1}^{d+1} \omega_j=1$. Assume that $\omega_i \le 1/d$ for every $i\in[d+1]$. Then there exists a hyperplane $h$ such that each open halfspace $H$ defined by $h$ satisfies $\mu_i(H) \le (\sum_{j=1}^{d+1} \mu_j(H))/d$ for every $i \in [d+1]$ and $\sum_{j=1}^{d+1} \mu_j(H) \ge \min(1/2, 1-d\omega) \ge 1/(d+1)$. As a consequence we obtain that every $(d+1)$-colored set of $nd$ points in $\mathbb{R}^d$ such that no color is used for more than $n$ points can be partitioned into $n$ disjoint rainbow $(d-1)$-dimensional simplices.Comment: 10 pages, 2 figures; corrected a serious error in the proof of Theorem 8, further minor change

“…There is plentiful research on various geometric problems involving pairings without crossings. Some of considered problems examine matchings of various planar objects, see [7,6,13], while more basic problems involve matching pairs of points by straight line segments, see [4,3,5]. There is always a non-crossing matching of points with non-crossing segments, and moreover it is straightforward to prove that a matching minimizing the total sum of lengths of its segments has to be non-crossing.…”

Section: Related Workmentioning

confidence: 99%

Faster bottleneck non-crossing matchings of points in convex position

Savić

Stojaković

2017

Given an even number of points in a plane, we are interested in matching all the points by straight line segments so that the segments do not cross. Bottleneck matching is a matching that minimizes the length of the longest segment. For points in convex position, we present a quadratic-time algorithm for finding a bottleneck non-crossing matching, improving upon the best previously known algorithm of cubic time complexity.