On balanced 4-holes in bichromatic point sets

by S. Bereg, J. M. Díaz-Báñez, R. Fabila-Monroy, P. Pérez-Lantero, A. Ramírez-Vigueras, T. Sakai, J. Urrutia, I. Ventura

Abstract: Let S=R ∪ B be a point set in the plane in general position such that each of its elements is colored either red or blue, where R and B denote the points colored red and the points colored blue, respectively. A quadrilateral with vertices in S is called a 4-hole if its interior is empty of elements of S. We say that a 4-hole of S is balanced if it has 2 red and 2 blue points of S as vertices. In this paper, we prove that if R and B contain n points each then S has at least (n2-4n)/12 balanced 4-holes, and this bound is tight up to a constant factor. Since there are two-colored point sets with no balanced convex 4-holes, we further provide a characterization of the two-colored point sets having this type of 4-holes.

A prior version has been presented at the 28th European Workshop on Computational Geometry (EuroCG 2012).
Example: Balanced 4-holes in a set of red/blue points.

pdf file submitted to the journal.