论文标题
紧密的汉密尔顿周期的局部代码条件3-均匀的超图
Localised codegree conditions for tight Hamilton cycles in 3-uniform hypergraphs
论文作者
论文摘要
我们研究了足够的条件，以统一密集的$ 3 $均匀的超图。 Lenz，Mubayi和Mycroft首先考虑了这类类型的问题，以解决汉密尔顿周期，Aigner-Horev和Levy认为这是汉密尔顿周期紧密的汉密尔顿周期的，这是一个相当强烈的统一密集的超透明牌的概念。我们专注于紧密的循环，并获得较弱的均匀致密性超图概念的最佳结果。 我们表明，如果$ n $ vertex $ 3 $ - 均匀的超graph $ h =（v，e）$具有任何属性，对于任何一套顶点$ x $，对于任何集合$ p $ p $的顶点，由$ p $和一个来自$ x $ x $ $ x $的$ p $组成的超计划数量至少是$ x $（至少是$ x $（$ x $） -o（| v |^3）$和$ h $的最低顶点学位至少$ω（| v |^2）$，然后$ h $包含一个紧密的汉密尔顿周期。概率的结构表明，在这种情况下，常数$ 1/4 $是最佳的。
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.
