Rainbow polygons for colored point sets in the plane
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arXiv
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| Format: | Preprint |
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2020
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| author | Flores-Peñaloza, David Kano, Mikio Martínez-Sandoval, Leonardo Orden, David Tejel, Javier Tóth, Csaba D. Urrutia, Jorge Vogtenhuber, Birgit |
| author_facet | Flores-Peñaloza, David Kano, Mikio Martínez-Sandoval, Leonardo Orden, David Tejel, Javier Tóth, Csaba D. Urrutia, Jorge Vogtenhuber, Birgit |
| contents | 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 $\operatorname{rb-index}(S)$ denote the smallest size of a perfect rainbow polygon for a colored point set $S$, and let $\operatorname{rb-index}(k)$ be the maximum of $\operatorname{rb-index}(S)$ over all $k$-colored point sets in general position; that is, every $k$-colored point set $S$ has a perfect rainbow polygon with at most $\operatorname{rb-index}(k)$ vertices. In this paper, we determine the values of $\operatorname{rb-index}(k)$ up to $k=7$, which is the first case where $\operatorname{rb-index}(k)\neq k$, and we prove that for $k\ge 5$, \[ \frac{40\lfloor (k-1)/2 \rfloor -8}{19} %Birgit: \leq\operatorname{rb-index}(k)\leq 10 \bigg\lfloor\frac{k}{7}\bigg\rfloor + 11. \] Furthermore, for a $k$-colored set of $n$ points in the plane in general position, a perfect rainbow polygon with at most $10 \lfloor\frac{k}{7}\rfloor + 11$ vertices can be computed in $O(n\log n)$ time. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2007_10139 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Rainbow polygons for colored point sets in the plane Flores-Peñaloza, David Kano, Mikio Martínez-Sandoval, Leonardo Orden, David Tejel, Javier Tóth, Csaba D. Urrutia, Jorge Vogtenhuber, Birgit Computational Geometry Discrete Mathematics Combinatorics 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 $\operatorname{rb-index}(S)$ denote the smallest size of a perfect rainbow polygon for a colored point set $S$, and let $\operatorname{rb-index}(k)$ be the maximum of $\operatorname{rb-index}(S)$ over all $k$-colored point sets in general position; that is, every $k$-colored point set $S$ has a perfect rainbow polygon with at most $\operatorname{rb-index}(k)$ vertices. In this paper, we determine the values of $\operatorname{rb-index}(k)$ up to $k=7$, which is the first case where $\operatorname{rb-index}(k)\neq k$, and we prove that for $k\ge 5$, \[ \frac{40\lfloor (k-1)/2 \rfloor -8}{19} %Birgit: \leq\operatorname{rb-index}(k)\leq 10 \bigg\lfloor\frac{k}{7}\bigg\rfloor + 11. \] Furthermore, for a $k$-colored set of $n$ points in the plane in general position, a perfect rainbow polygon with at most $10 \lfloor\frac{k}{7}\rfloor + 11$ vertices can be computed in $O(n\log n)$ time. |
| title | Rainbow polygons for colored point sets in the plane |
| topic | Computational Geometry Discrete Mathematics Combinatorics |
| url | https://arxiv.org/abs/2007.10139 |