Rainbow polygons for colored point sets in the plane
David Flores-Pe\~naloza, Mikio Kano, Leonardo Mart\'inez-Sandoval,, David Orden, Javier Tejel, Csaba D. T\'oth, Jorge Urrutia, Birgit Vogtenhuber

TL;DR
This paper studies the minimal size of polygons that contain exactly one point of each color in a colored point set, determining bounds for the maximum such size for up to seven colors and providing an efficient construction algorithm.
Contribution
It determines the exact values of the rainbow polygon index for up to 7 colors and establishes bounds for larger numbers, along with an efficient algorithm for constructing such polygons.
Findings
Exact rainbow polygon index for up to 7 colors
Bounds for rainbow polygon index for larger color sets
Efficient algorithm for constructing perfect rainbow polygons
Abstract
Given a colored point set in the plane, a perfect rainbow polygon is a simple polygon that contains exactly one point of each color, either in its interior or on its boundary. Let denote the smallest size of a perfect rainbow polygon for a colored point set , and let be the maximum of over all -colored point sets in general position; that is, every -colored point set has a perfect rainbow polygon with at most vertices. In this paper, we determine the values of up to , which is the first case where , and we prove that for , \[ \frac{40\lfloor (k-1)/2 \rfloor -8}{19} %Birgit: \leq\operatorname{rb-index}(k)\leq 10 \bigg\lfloor\frac{k}{7}\bigg\rfloor + 11. \] Furthermore, for a -colored…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
