  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.
