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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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