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Disproving Two Conjectures on the Hamiltonicity of Venn Diagrams

Authors Sofia Brenner , Linda Kleist , Torsten Mütze , Christian Rieck , Francesco Verciani

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Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Mathematics of computing → Discrete mathematics
Keywords
  • Venn diagram
  • Winkler’s conjecture
  • Hamilton cycle
  • perfect matching
  • hypercube

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Abstract

In 1984, Winkler conjectured that every simple Venn diagram with n curves can be extended to a simple Venn diagram with n+1 curves. This conjecture is equivalent to the statement that the dual graph of any simple Venn diagram has a Hamilton cycle. In this work, we construct counterexamples to Winkler’s conjecture for all n ≥ 6. As part of this proof, we computed all 3.430.404 simple Venn diagrams with n = 6 curves (even their number was not previously known), among which we found 72 counterexamples.
We also disprove another conjecture about the Hamiltonicity of the arrangement graph of a Venn diagram. Specifically, while working on Winkler’s conjecture, Pruesse and Ruskey proved that this graph has a Hamilton cycle for every simple Venn diagram with n curves, and conjectured that this also holds for non-simple diagrams. We construct counterexamples to this conjecture for all n ≥ 4.

Cite As Get BibTex

Sofia Brenner, Linda Kleist, Torsten Mütze, Christian Rieck, and Francesco Verciani. Disproving Two Conjectures on the Hamiltonicity of Venn Diagrams. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 22:1-22:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.SoCG.2026.22

Author Details

Sofia Brenner
  • Institute of Mathematics, University of Kassel, Germany
Linda Kleist
  • Department of Informatics, University of Hamburg, Germany
Torsten Mütze
  • Institute of Mathematics, University of Kassel, Germany
Christian Rieck
  • Institute of Mathematics, University of Kassel, Germany
Francesco Verciani
  • Institute of Mathematics, University of Kassel, Germany

Funding

Sofia Brenner, Torsten Mütze, Christian Rieck, and Francesco Verciani were supported by German Science Foundation (DFG) grant 522790373. Sofia Brenner also acknowledges funding by a postdoc fellowship of the German Academic Exchange Service (DAAD).

Supplementary Materials

References

  1. Sofia Brenner, Linda Kleist, Torsten Mütze, Christian Rieck, and Francesco Verciani. Disproving two conjectures on the Hamiltonicity of Venn diagrams, 2025. URL: https://arxiv.org/abs/2503.18554.
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    Software Heritage Logo archived version
    full metadata available at: https://doi.org/10.4230/artifacts.26092
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