## Rainbow polygons for colored point sets in the plane.(English)Zbl 1466.52007

Summary: 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 $$k \geq 5$$, $\frac{ 40 \lfloor ( k - 1 ) / 2 \rfloor - 8}{ 19} \leq \operatorname{rb-index} ( k ) \leq 10 \left\lfloor \frac{ k}{ 7} \right\rfloor + 11 .$ 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.

### MSC:

 52A37 Other problems of combinatorial convexity 52C10 Erdős problems and related topics of discrete geometry 52A10 Convex sets in $$2$$ dimensions (including convex curves)
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### References:

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