Intersecting diametral balls induced by a geometric graph

TL;DR

This paper explores conditions under which certain geometric graphs, constructed from points in Euclidean space, have intersecting balls induced by their edges, revealing new Tverberg graph properties in planar and higher dimensions.

Contribution

It proves the existence of Hamiltonian cycles and perfect matchings that are Tverberg graphs in both planar and Euclidean spaces, extending Tverberg-type intersection results.

Findings

Existence of Hamiltonian Tverberg cycles in planar point sets

Existence of perfect red-blue Tverberg matchings in the plane

Existence of perfect open Tverberg matchings in Euclidean space

Abstract

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.