On a Tverberg graph

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