Intersecting Diametral Balls Induced by a Geometric Graph

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