Disproving two conjectures on the Hamiltonicity of Venn diagrams
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arXiv
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2025
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| author | Brenner, Sofia Kleist, Linda Mütze, Torsten Rieck, Christian Verciani, Francesco |
| author_facet | Brenner, Sofia Kleist, Linda Mütze, Torsten Rieck, Christian Verciani, Francesco |
| contents | In 1984, Winkler conjectured that every simple Venn diagram with $n$ curves can be extended to a simple Venn diagram with $n+1$ curves. His 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 $n\geq 6$. 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 construct monotone Venn diagrams, i.e., diagrams that can be drawn with $n$ convex curves, and are not extendable, for all $n\geq 7$. Furthermore, we also disprove another conjecture about the Hamiltonicity of the (primal) 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 $n\geq 4$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_18554 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Disproving two conjectures on the Hamiltonicity of Venn diagrams Brenner, Sofia Kleist, Linda Mütze, Torsten Rieck, Christian Verciani, Francesco Combinatorics Discrete Mathematics In 1984, Winkler conjectured that every simple Venn diagram with $n$ curves can be extended to a simple Venn diagram with $n+1$ curves. His 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 $n\geq 6$. 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 construct monotone Venn diagrams, i.e., diagrams that can be drawn with $n$ convex curves, and are not extendable, for all $n\geq 7$. Furthermore, we also disprove another conjecture about the Hamiltonicity of the (primal) 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 $n\geq 4$. |
| title | Disproving two conjectures on the Hamiltonicity of Venn diagrams |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2503.18554 |