# Tverberg’s Theorem, Disks, and Hamiltonian Cycles

@article{Soberon2021TverbergsTD, title={Tverberg’s Theorem, Disks, and Hamiltonian Cycles}, author={Pablo Sober'on and Yaqian Tang}, journal={Annals of Combinatorics}, year={2021} }

For a finite set of $S$ points in the plane and a graph with vertices on $S$ consider the disks with diameters induced by the edges. We show that for any odd set $S$ there exists a Hamiltonian cycle for which these disks share a point, and for an even set $S$ there exists a Hamiltonian path with the same property. We discuss high-dimensional versions of these theorems and their relation to other results in discrete geometry.

## 4 Citations

### Tverberg ’ s theorem is one of the essential results of modern discrete and convex geom

- Mathematics
- 2022

For a graph whose vertex set is a finite set of points in the Euclidean dspace consider the closed (open) balls with diameters induced by its edges. The graph is called a (an open) Tverberg graph if…

### COUNTEREXAMPLES TO THE COLORFUL TVERBERG CONJECTURE FOR HYPERPLANES

- MathematicsActa Mathematica Hungarica
- 2022

Karasev [16] conjectured that for every set of r blue lines, r green lines, and r red lines in the plane, there exists a partition of them into r colorful triples whose induced triangles intersect.…

### On Maximum-Sum Matchings of Points

- MathematicsJournal of Global Optimization
- 2022

It is proved that in this case all disks of the matching do have a common point, which implies a big improvement on a conjecture of Andy Fingerhut in 1995, about a maximum matching of $2n$ points in the plane.

### On a Tverberg graph

- MathematicsArXiv
- 2021

Using the idea of halving lines, it is shown that for any finite set of points in the plane, there exists a Hamiltonian cycle that is a Tverberg graph and for any n red and n blue points in R, there is a perfect red-blue matching that is an open TVerberg graph.

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