For positive integers r >ℓ , an r ‐uniform hypergraph is called an ℓ ‐cycle if there exists a cyclic ordering of its vertices such that each of its edges consists of r consecutive vertices, and such that every pair of consecutive edges (in the natural ordering of the edges) intersect in precisely ℓ vertices; such cycles are said to be linear when ℓ =1, and nonlinear when ℓ >1. We determine the sharp threshold for nonlinear Hamiltonian cycles and show that for all r >ℓ >1, the threshold for the appearance of a Hamiltonian ℓ ‐cycle in the random r ‐uniform hypergraph on n vertices is sharp and given by for an explicitly specified function λ . This resolves several questions raised by Dudek and Frieze in 2011.10

中文翻译：

---

曲线图中非线性哈密顿循环的尖锐阈值

为正整数ř > ℓ，一个ř -uniform超图被称为ℓ如果存在其顶点的环状排序，使得其各个边缘的由-cycle ř连续的顶点，并且使得每对连续的边缘的（在边缘的自然顺序）恰好相交于ℓ个顶点；这种循环被认为是线性时ℓ = 1，且当非线性ℓ > 1。我们确定了非线性哈密顿循环的尖锐阈值，并表明对于所有r > ℓ > 1，哈密​​顿ian出现的阈值在n个顶点上的随机r均匀超图中的循环是尖锐的，并且由明确指定的函数λ给出。这解决了Dudek和Frieze在2011.10中提出的几个问题。