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 $n\geq 4$.

翻译：1984年，Winkler猜想：任何具有$n$条曲线的简单维恩图均可扩展为具有$n+1$条曲线的简单维恩图。该猜想等价于断言任何简单维恩图的对偶图均包含哈密顿环。本文针对所有$n\geq 6$的情形构造了Winkler猜想的反例。作为证明的一部分，我们计算了全部3,430,404个具有$n=6$条曲线的简单维恩图（此前甚至该数量亦属未知），并从中发现72个反例。同时，我们还对所有$n\geq 7$的情形构造了不可扩展的单调维恩图（即可以用$n$条凸曲线绘制的图）。此外，我们还证伪了关于维恩图（原图）哈密顿性的另一猜想。具体而言，Pruesse与Ruskey在研究Winkler猜想时证明了：对于任意具有$n$条曲线的简单维恩图，其原图均包含哈密顿环，并推测该性质对非简单维恩图同样成立。我们针对所有$n\geq 4$的情形构造了该猜想的反例。