On a Tverberg graph

08/22/2021

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by   Olimjoni Pirahmad, et al.

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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. Also, 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.