Intersecting diametral balls induced by a geometric graph

For a graph whose vertex set is a finite set of points in the Euclidean $d$-space 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. Also, we prove that ($iii$) for any even set of points in the Euclidean $d$-space, there exists a perfect matching that is an open Tverberg graph; ($iv$) for any $n$ red and $n$ blue points in the Euclidean $d$-space, there exists a perfect red-blue matching that is a Tverberg graph.