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.
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The authors would like to thank the anonymous referee for the detailedcomments on this manuscript.
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This research project was done as part of the 2020 Baruch Discrete Mathematics REU, supported by NSF awards DMS-1802059, DMS-1851420, and DMS-1953141. Soberón’s research is also supported by PSC-CUNY grant 63529-00 51. Tang’s research was supported by Wesleyan University’s Summer Science Research Endowed Fund.
Communicated by Victor Chepoi.
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Soberón, P., Tang, Y. Tverberg’s Theorem, Disks, and Hamiltonian Cycles. Ann. Comb. (2021). https://doi.org/10.1007/s00026-021-00557-0