In this note we study the emergence of Hamiltonian Berge cycles in random r-uniform hypergraphs. For $r\geq 3$ we prove an optimal stopping time result that if edges are sequentially added to an initially empty r-graph, then as soon as the minimum degree is at least 2, the hypergraph with high probability has such a cycle. In particular, this determines the threshold probability for Berge Hamiltonicity of the Erdős–Rényi random r-graph, and we also show that the 2-out random r-graph with high probability has such a cycle. We obtain similar results for weak Berge cycles as well, thus resolving a conjecture of Poole.

中文翻译：

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随机超图中的哈密顿伯杰循环

在这篇笔记中，我们研究了随机中哈密顿伯奇循环的出现r-统一的超图。为了$r\geq 3$我们证明了一个最优的停止时间结果，如果边被顺序添加到一个最初为空的r-graph，那么只要最小度数至少为 2，那么大概率的超图就有这样的循环。特别是，这决定了 Erdős–Rényi 随机数的 Berge 哈密顿性的阈值概率r-graph，我们也证明了2- 随机输出r- 高概率的图有这样一个循环。我们得到了类似的结果弱贝尔格循环，从而解决了普尔猜想。