On Maximum-Sum Matchings of Points

Abstract

Huemer et al. (Discrete Mathematics, 2019) proved that for any two point sets R and B with |R| = |B|, the perfect matching that matches points of R with points of B, and maximizes the total squared Euclidean distance of the matched pairs, verifies that all the disks induced by the matching have a common point. Each pair of matched points p ∈ R and q ∈ B induces the disk of smallest diameter that covers p and q. Following this research line, in this paper we consider the perfect matching that maximizes the total Euclidean distance. First, we prove that this new matching for R and B does not always ensure the common intersection property of the disks. Second, we extend the study of this new matching for sets of 2n uncolored points in the plane, where a matching is just a partition of the points into n pairs. As the main result, we prove that in this case all disks of the matching do have a common point. This implies a big improvement on a conjecture of Andy Fingerhut in 1995, about a maximum matching of 2n points in the plane.

Intersecting diametral balls induced by a geometric graph

“…It seems that the mentioned authors overlooked Lemma 2 from the paper [3] of Dumitrescu, Pach, and Tóth. This lemma easily implies the following strengthening of the result from [2]: Given 2n distinct points in the plane, one can denote them as x 1 , . .…”

“…We say that G is a Tverberg graph if In 2019, Huemer, Pérez-Lantero, Seara, and Silveira [5] showed that for any n red points and any n blue points in the plane, there is a red-blue Tverberg matching (every edge of this Tverberg matching connects a red vertex with a blue one). Later, Bereg, Chacón-Rivera, Flores-Peñaloza, Huemer, and Pérez-Lantero [2] found a second proof of the monochromatic version of this result, that is, for any 2n points in the plane, there is a Tverberg matching. Recently, Soberón and Tang [8] showed the existence of a Tverberg cycle for an odd set of points in the plane.…”

“…It claims that for any set of (r − 1)(d + 1) + 1 points in R d , there exists a partition of points into r parts whose convex hulls intersect. In the current paper, we consider a variation of Tverberg's problem introduced recently in [2,5,8].…”