3 Search Results for "Hamburger, Peter"

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DOI: 10.4230/LIPIcs.SoCG.2026.22

Disproving Two Conjectures on the Hamiltonicity of Venn Diagrams

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

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

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

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)

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@InProceedings{brenner_et_al:LIPIcs.SoCG.2026.22,
  author =	{Brenner, Sofia and Kleist, Linda and M\"{u}tze, Torsten and Rieck, Christian and Verciani, Francesco},
  title =	{{Disproving Two Conjectures on the Hamiltonicity of Venn Diagrams}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{22:1--22:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.22},
  URN =		{urn:nbn:de:0030-drops-258285},
  doi =		{10.4230/LIPIcs.SoCG.2026.22},
  annote =	{Keywords: Venn diagram, Winkler’s conjecture, Hamilton cycle, perfect matching, hypercube}
}

Document

DOI: 10.4230/LIPIcs.SoCG.2026.21

On Minimum Venn Diagrams

Authors: Sofia Brenner, Petr Gregor, Torsten Mütze, and Francesco Verciani

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract

An n-Venn diagram is a diagram in the plane consisting of n simple closed curves that intersect only finitely many times such that each of the 2ⁿ possible intersections of their interiors is represented by a single connected region. An n-Venn diagram has at most 2ⁿ-2 crossings, and if this maximum number of crossings is attained, then only two curves intersect in every crossing. To complement this, Bultena and Ruskey considered n-Venn diagrams that minimize the number of crossings, which implies that many curves intersect in every crossing. Specifically, they proved that the total number of crossings in any n-Venn diagram is at least L_n≔⌈(2ⁿ-2)/(n-1)⌉, and if this lower bound is attained, then essentially all n curves intersect in every crossing. Diagrams achieving this bound are called minimum Venn diagrams, and are known only for n ≤ 7. Bultena and Ruskey conjectured that they exist for all n ≥ 8. In this work, we establish an asymptotic version of their conjecture. For n = 8 we construct a diagram with 40 crossings, only 3 more than the lower bound L₈ = 37. Furthermore, for every n of the form n = 2^k for some integer k ≥ 4, we construct an n-Venn diagram with at most (1+33/8n)L_n = (1+o(1))L_n many crossings. Via a doubling trick this also gives (n+m)-Venn diagrams for all 0 ≤ m < n with at most 40⋅ 2^m crossings for n = 8 and at most (1+33/8n) (n+m)/n L_{n+m} = (2+o(1))L_{n+m} many crossings for k ≥ 4. In particular, we obtain n-Venn diagrams with the smallest known number of crossings for all n ≥ 8. Our constructions are based on partitions of the hypercube into isometric paths and cycles, using a result of Ramras.

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Sofia Brenner, Petr Gregor, Torsten Mütze, and Francesco Verciani. On Minimum Venn Diagrams. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 21:1-21:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{brenner_et_al:LIPIcs.SoCG.2026.21,
  author =	{Brenner, Sofia and Gregor, Petr and M\"{u}tze, Torsten and Verciani, Francesco},
  title =	{{On Minimum Venn Diagrams}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{21:1--21:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.21},
  URN =		{urn:nbn:de:0030-drops-258278},
  doi =		{10.4230/LIPIcs.SoCG.2026.21},
  annote =	{Keywords: Venn diagram, crossing, conjecture, hypercube, partition}
}

Document

DOI: 10.4230/LIPIcs.MFCS.2018.69

Double Threshold Digraphs

Authors: Peter Hamburger, Ross M. McConnell, Attila Pór, Jeremy P. Spinrad, and Zhisheng Xu

Published in: LIPIcs, Volume 117, 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)

Abstract

A semiorder is a model of preference relations where each element x is associated with a utility value alpha(x), and there is a threshold t such that y is preferred to x iff alpha(y) - alpha(x) > t. These are motivated by the notion that there is some uncertainty in the utility values we assign an object or that a subject may be unable to distinguish a preference between objects whose values are close. However, they fail to model the well-known phenomenon that preferences are not always transitive. Also, if we are uncertain of the utility values, it is not logical that preference is determined absolutely by a comparison of them with an exact threshold. We propose a new model in which there are two thresholds, t_1 and t_2; if the difference alpha(y) - alpha(x) is less than t_1, then y is not preferred to x; if the difference is greater than t_2 then y is preferred to x; if it is between t_1 and t_2, then y may or may not be preferred to x. We call such a relation a (t_1,t_2) double-threshold semiorder, and the corresponding directed graph G = (V,E) a (t_1,t_2) double-threshold digraph. Every directed acyclic graph is a double-threshold digraph; increasing bounds on t_2/t_1 give a nested hierarchy of subclasses of the directed acyclic graphs. In this paper we characterize the subclasses in terms of forbidden subgraphs, and give algorithms for finding an assignment of utility values that explains the relation in terms of a given (t_1,t_2) or else produces a forbidden subgraph, and finding the minimum value lambda of t_2/t_1 that is satisfiable for a given directed acyclic graph. We show that lambda gives a useful measure of the complexity of a directed acyclic graph with respect to several optimization problems that are NP-hard on arbitrary directed acyclic graphs.

Cite as

Peter Hamburger, Ross M. McConnell, Attila Pór, Jeremy P. Spinrad, and Zhisheng Xu. Double Threshold Digraphs. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 69:1-69:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{hamburger_et_al:LIPIcs.MFCS.2018.69,
  author =	{Hamburger, Peter and McConnell, Ross M. and P\'{o}r, Attila and Spinrad, Jeremy P. and Xu, Zhisheng},
  title =	{{Double Threshold Digraphs}},
  booktitle =	{43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)},
  pages =	{69:1--69:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-086-6},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{117},
  editor =	{Potapov, Igor and Spirakis, Paul and Worrell, James},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2018.69},
  URN =		{urn:nbn:de:0030-drops-96519},
  doi =		{10.4230/LIPIcs.MFCS.2018.69},
  annote =	{Keywords: posets, preference relations, approximation algorithms}
}

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3 Search Results for "Hamburger, Peter"

Disproving Two Conjectures on the Hamiltonicity of Venn Diagrams

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On Minimum Venn Diagrams

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Double Threshold Digraphs

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