All Simple Venn Diagrams are Hamiltonian

TL;DR

The paper proves that all simple Venn diagrams, modeled as certain 4-connected graphs, are Hamiltonian, extending the understanding of their combinatorial structure and intersection properties.

Contribution

It establishes that connected collections of simple closed curves with specific intersection properties are 4-connected and Hamiltonian, generalizing previous results on Venn diagrams.

Findings

All simple Venn diagrams are Hamiltonian.

Connected collections of such curves are 4-connected.

These properties hold under transversal intersections with at most two curves intersecting at a point.

Abstract

An $n$-Venn diagram is a certain collection of $n$ simple closed curves in the plane. They can be regarded as graphs where the points of intersection are vertices and the curve segments between points of intersection are edges. Every $n$-Venn diagram has the property that a curve touches any given face at most once between the points of intersection incident to that face. We prove that any connected collection of $n$ simple closed curves satisfying that property are 4-connected, if $n \ge 3$, so long as the curves intersect transversally and at most two curves intersect at any point. Hence by a theorem of Tutte, such collections, including simple Venn diagrams, are Hamiltonian.