On Maximum-Sum Matchings of Points
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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.
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