Intersecting diametral balls induced by a geometric graph

Abstract

For a graph whose vertex set is a finite set of points in R d , consider the closed (open) balls with diameters induced by its edges. The graph is called a (an open) Tverberg graph if these closed (open) balls intersect. Using the idea of halving lines, we show that (i) for any finite set of points in the plane, there exists a Hamiltonian cycle that is a Tverberg graph; (ii) for any n red and n blue points in the plane, there exists a perfect red-blue matching that is a Tverberg graph. Using the idea of infinite descent, we prove that (iii) for any even set of points in R d , there exists a perfect matching that is an open Tverberg graph; (iv ) for any n red and n blue points in R d , there exists a perfect red-blue matching that is a Tverberg graph.

On maximum-sum matchings of points

“…Concerning disks with a common intersection, Soberón and Tang [22] proved recently that if G is the complete geometric graph on n points in the plane, there exists a Hamiltonian cycle (or path) H in G such that all the disks D e , where e is an edge of H , have a common intersection. Recently, Pirahmad et al [21] have extended the results of Huemer et al [18]. They gave an alternative proof that for any n red and n blue points in the plane, there exists a perfect red-blue matching such that the disks associated with the matching have a common intersection.…”