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

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COUNTEREXAMPLES TO THE COLORFUL TVERBERG CONJECTURE FOR HYPERPLANES

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

TLDR
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.

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TLDR
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