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Disproving two conjectures on the Hamiltonicity of Venn diagrams

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

TL;DR

The paper resolves two longstanding conjectures about Hamiltonicity in Venn diagrams by constructing explicit counterexamples: for every $n\ge6$ there exist simple $n$-Venn quadrangulations without a Hamilton cycle, and for every $n\ge4$ non-simple $n$-Venn diagrams whose primal graphs lack a Hamilton cycle. The authors introduce a ladder-based extension lemma that preserves local obstructions, enabling an inductive construction of nonextendable diagrams and their monotone variants for growing $n$. A central technical achievement is a full computer-assisted census of all simple $6$-Venn quadrangulations, performed via dynamic programming on two halves of the diagram, with detailed steps to enumerate boundary cycles, fill interiors with 4-faces, pair compatible halves, remove isomorphs, and analyze graph properties, yielding precise counts such as $3{,}430{,}404$ diagrams and $72$ that lack a Hamilton cycle. These results demonstrate concrete structural obstructions to extending Venn diagrams and provide comprehensive data guiding future combinatorial investigations into Venn diagram topology and related Hamiltonicity questions.

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. 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 $n\geq 4$.

Disproving two conjectures on the Hamiltonicity of Venn diagrams

TL;DR

The paper resolves two longstanding conjectures about Hamiltonicity in Venn diagrams by constructing explicit counterexamples: for every there exist simple -Venn quadrangulations without a Hamilton cycle, and for every non-simple -Venn diagrams whose primal graphs lack a Hamilton cycle. The authors introduce a ladder-based extension lemma that preserves local obstructions, enabling an inductive construction of nonextendable diagrams and their monotone variants for growing . A central technical achievement is a full computer-assisted census of all simple -Venn quadrangulations, performed via dynamic programming on two halves of the diagram, with detailed steps to enumerate boundary cycles, fill interiors with 4-faces, pair compatible halves, remove isomorphs, and analyze graph properties, yielding precise counts such as diagrams and that lack a Hamilton cycle. These results demonstrate concrete structural obstructions to extending Venn diagrams and provide comprehensive data guiding future combinatorial investigations into Venn diagram topology and related Hamiltonicity questions.

Abstract

In 1984, Winkler conjectured that every simple Venn diagram with curves can be extended to a simple Venn diagram with 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 . As part of this proof, we computed all 3.430.404 simple Venn diagrams with 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 convex curves, and are not extendable, for all . 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 curves, and conjectured that this also holds for non-simple diagrams. We construct counterexamples to this conjecture for all .

Paper Structure

This paper contains 16 sections, 6 theorems, 1 equation, 14 figures, 1 table.

Table of Contents

  1. Introduction
  2. Winkler's conjecture
  3. Translation to a Hamilton cycle problem
  4. The counterexamples
  5. Pruesse and Ruskey's conjecture
  6. The counterexamples
  7. Proof of Theorems \ref{['thm:counter1']} and \ref{['thm:counter1-monotone']}
  8. Proof of Theorem \ref{['thm:6']}
  9. Dynamic programming on two halves of a Venn diagram
  10. Step (i): Computing all cycles $C$ of $Q_{n-1}$
  11. Step (ii): Computing all $C$-quadrangulations
  12. Step (iii): Pairing up $C$-quadrangulations
  13. Step (iv): Filtering out isomorphic graphs
  14. Step (v): Analyzing properties of $n$-Venn quadrangulations
  15. Proof of Theorem \ref{['thm:counter2']}
  16. ...and 1 more sections

Key Result

Theorem 2

For every $n\geq 6$ there is an $n$-Venn quadrangulation that has no perfect matching and hence no Hamilton cycle. Its dual is a simple $n$-Venn diagram that cannot be extended to a simple $(n+1)$-Venn diagram by adding a suitable curve.

Figures (14)

  • Figure 1: Venn diagrams with $n=3,4,5$ curves: (a), (b), (c) are simple, whereas (d), (e), (f) are non-simple; (a), (b) and (d) are reducible, whereas (c), (e) and (f) are irreducible; (a), (b), (c) and (d) are monotone, whereas (e) and (f) are not monotone; (a), (b), (c), (d) and (f) are exposed, whereas (e) is not exposed.
  • Figure 2: Representation of the 5-Venn diagram from Figure \ref{['fig:345']} (c) as a wire diagram.
  • Figure 3: (a) A (monotone) 4-Venn diagram and its dual graph, a (monotone) 4-Venn quadrangulation; (b) a Hamilton cycle in the quadrangulation; (c) the 5-Venn diagram obtained from (a) by adding the curve corresponding to the Hamilton cycle in (b).
  • Figure 4: (Left) A counterexample to Winkler's conjecture for $n=6$, namely a simple (non-monotone) 6-Venn diagram that cannot be extended to a simple 7-Venn diagram by adding a suitable curve. Highlighted are 12 red regions that are surrounded by only 11 blue regions, which is a local obstruction for extendability. (Right) The dual graph of the diagram on the left, namely a 6-Venn quadrangulation $P$ that has no perfect matching and hence no Hamilton cycle. Edges are colored according to the colors of the curves. The highlighted subgraph $H$ contains a subset of 12 red vertices in one partition class and their 11 blue neighbors, witnessing the non-existence of a perfect matching. The green subgraph $L$ is the ladder used for extending the counterexample to all values of $n>6$, and all those extensions contain a copy of $H$.
  • Figure 5: A simple monotone 7-Venn diagram that cannot be extended to a simple 8-Venn diagram by adding a suitable curve. The bottom picture is the same diagram in a wire representation. The subgraph $H$ of the dual Venn quadrangulation is the same as in Figure \ref{['fig:counter1']}. It corresponds to the highlighted 12 red regions that are surrounded by only 11 blue regions, which is a local obstruction for extendability. The green subgraph $L$ is the ladder used for extending the counterexample to all values of $n>7$.
  • ...and 9 more figures

Theorems & Definitions (12)

  • Conjecture 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5: pruesse_ruskey_preprint
  • Conjecture 6: pruesse_ruskey_preprint
  • Theorem 7
  • Lemma 8
  • proof
  • proof : Proof of Theorem \ref{['thm:counter1']}
  • ...and 2 more