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 rb-index(S) denote the smallest size of a perfect rainbow polygon for a colored point set S, and let rb-index(k) be the maximum of 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 rb-index(k) vertices. In this paper, we determine the values of rb-index(k) up to k=7, which is the first case where rb-index(k)\neq k, and we prove that for k\ge 5, \[ 40\lfloor (k-1)/2 \rfloor -8{19} \%Birgit: \leqrb-index(k)\leq 10 \bigg\lfloork{7}\bigg\rfloor + 11. \] Furthermore, for a k-colored set of n points in the plane in general position, a perfect rainbow polygon with at most 10 \lfloork{7}\rfloor + 11 vertices can be computed in O(n\log n) time.
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 rb-index(S) denote the smallest size of a perfect rainbow polygon for a colored point set S, and let rb-index(k) be the maximum of 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 rb-index(k) vertices. In this paper, we determine the values of rb-index(k) up to k=7, which is the first case where rb-index(k)\neq k, and we prove that for k\ge 5, \[ 40\lfloor (k-1)/2 \rfloor -8{19} \%Birgit: \leqrb-index(k)\leq 10 \bigg\lfloork{7}\bigg\rfloor + 11. \] Furthermore, for a k-colored set of n points in the plane in general position, a perfect rainbow polygon with at most 10 \lfloork{7}\rfloor + 11 vertices can be computed in O(n\log n) time.
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