For integers $k\ge 3$ and $1\le \ell \le k-1$, we prove that for any $\alpha >0$, there exist $ϵ>0$ and $C>0$ such that for sufficiently large $n\in (k-\ell )\mathbb{N}$, the union of a k-uniform hypergraph with minimum vertex degree $\alpha n^{k-1}$ and a binomial random k-uniform hypergraph ${\mathbb{G}}^{(k)}(n,p)$ with $p\ge n^{-(k-\ell )-ϵ}$ for $\ell \ge 2$ and $p\ge C{n}^{-(k-1)}$ for $\ell =1$ on the same vertex set contains a Hamiltonian ℓ-cycle with high probability. Our result is best possible up to the values of ϵ and C and answers a question of Krivelevich, Kwan and Sudakov.

中文翻译：

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随机摄动超图中的汉密尔顿性

对于整数 $ķ\ge 3$ 和 $\mathrm{1个}\le \ell \le ķ-\mathrm{1个}$，我们证明 $\alpha >0$， 存在 $ϵ>0$ 和 $C>0$ 这样足够大 $ñ\in （ķ-\ell ）\mathbb{ñ}$，具有最小顶点度的k一致超图的并集$\alpha {ñ}^{ķ-\mathrm{1个}}$和二项式随机k一致超图${\mathbb{G}}^{（ķ）}（ñ，p）$ 与 $p\ge {ñ}^{-（ķ-\ell ）-ϵ}$ 对于 $\ell \ge 2$ 和 $p\ge C{ñ}^{-（ķ-\mathrm{1个}）}$ 对于 $\ell =\mathrm{1个}$在同一顶点集上的汉密尔顿cycle-周期概率很高。我们的结果最可能达到ϵ和C的值，并回答了Krivelevich，Kwan和Sudakov的问题。