Rainbow Polygons For Colored Point Sets In The Plane

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) k not equal k, and we prove that for k >= 5,40 right perpendicular(k-1)/2 left perpendicular - 8/19 <= rb-index(k) <= right perpendicular k/7 left perpendicular + 11.Furthermore, for a k-colored set of n points sn the plane in general position, a perfect rainbow polygon with at most 10 right perpendicular k/7 left perpendicular + 11 vertices can be computed in O(n log n) time. (C) 2021 Elsevier B.V. All rights reserved.