AI 中文总结
该研究确定了一致稠密3-均匀超图中紧哈密顿圈的最小代码度条件，证明1/3为渐近最优常数，同时否定回答了相关问题并明确顶点度阈值未因一致稠密性降低。
AI 中文摘要
我们研究一致稠密3-均匀超图中紧哈密顿圈的最小度条件。证明对任意d,α>0，每个足够大的n顶点(ρ,d)-稠密3-均匀超图，若最小代码度至少为(1/3+α)n，则包含紧哈密顿圈，这以更强形式解决了Aigner-Horev与Levy的问题，且常数1/3是渐近最优的。还表明一致稠密性不会降低渐近顶点度阈值：存在(ρ,d)-稠密3-均匀超图，其最小顶点度为(5/9-o(1))C(n,2)且无紧哈密顿圈。最后构造了(ρ,2−√3)-稠密的例子，其最小代码度为(2−√3−o(1))n且无紧哈密顿圈，否定回答了Araújo、Piga与Schacht的问题。
英文摘要
We study minimum degree conditions for tight Hamiltonian cycles in uniformly dense $3$-uniform hypergraphs. We prove that for every $d,α>0$, every sufficiently large $(ρ,d)$-dense $3$-graph on $n$ vertices with minimum codegree at least $(1/3+α)n$ contains a tight Hamiltonian cycle. This resolves a problem of Aigner-Horev and Levy in a stronger form, and the constant $1/3$ is asymptotically best possible. We also show that uniform density does not lower the asymptotic vertex-degree threshold: there are $(ρ,d)$-dense $3$-graphs with minimum vertex degree $(5/9-o(1))\binom{n}{2}$ and no tight Hamiltonian cycle. Finally, we construct $(ρ,2-\sqrt{3})$-dense examples with minimum codegree $(2-\sqrt{3}-o(1))n$ and no tight Hamiltonian cycle, answering negatively a question of Ara{ú}jo, Piga and Schacht.
Comments43 pages, including a 7-page appendix, 3 figures
