3-一致超图中紧Hamilton圈的局部余度条件
Localised codegree conditions for tight Hamilton cycles in 3-uniform hypergraphs
摘要：
研究了一致稠密$3$ -一致超图中Hamilton圈存在的充分条件。这种类型的问题首先被Lenz、Mubayi和Mycroft考虑为松散的Hamilton圈，而Aigner-Horev和Levy则考虑为紧致的Hamilton圈，并给出了一个相当强的一致稠密超图的概念。我们专注于紧圈，并获得最佳结果的一个较弱的概念一致稠密超图。本文证明了：如果$n$ -顶点$3$ -一致超图$H=(V,E)$具有这样的性质：对于任意顶点集$X$和任意顶点对的集合$P$，属于$P$的顶点对和来自$X$的顶点组成的超边的个数至少为$(1/4+o(1))|X||P| - o(|V|^3)$，且$H$的最小顶点度至少为$Ω(|V|^2)$，则$H$包含紧Hamilton圈.概率构造表明，常数$1/4$在这种情况下是最佳的。
英文摘要：
We study sufficient conditions for the existence of Hamilton cycles in uniformly dense $3$-uniform hypergraphs. Problems of this type were first considered by Lenz, Mubayi, and Mycroft for loose Hamilton cycles and Aigner-Horev and Levy considered it for tight Hamilton cycles for a fairly strong notion of uniformly dense hypergraphs. We focus on tight cycles and obtain optimal results for a weaker notion of uniformly dense hypergraphs. We show that if an $n$-vertex $3$-uniform hypergraph $H=(V,E)$ has the property that for any set of vertices $X$ and for any collection $P$ of pairs of vertices, the number of hyperedges composed by a pair belonging to $P$ and one vertex from $X$ is at least $(1/4+o(1))|X||P| - o(|V|^3)$ and $H$ has minimum vertex degree at least $Ω(|V|^2)$, then $H$ contains a tight Hamilton cycle. A probabilistic construction shows that the constant $1/4$ is optimal in this context.
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