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All Simple Venn Diagrams are Hamiltonian

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.