论文标题
紧密的汉密尔顿周期的最低度条件
Minimum degree conditions for tight Hamilton cycles
论文作者
论文摘要
我们开发了一个新的框架，以研究$ k $均匀的超图中的最低$ d $数量条件，以保证存在紧密的汉密尔顿周期。我们的主要理论结果涉及一次典型的吸收，路径覆盖和连接所有$ K $和$ d $的参数，因此阐明了基本的结构问题。在此基础上，我们表明，可以通过关注社区的内部结构来研究$ k $均匀的汉密尔顿周期的最低$ d $数度条件。这将问题降低到了$（k-d）$ - 统一超图（gallai type）的问题，这是一个独立的兴趣。 建立此框架后，我们可以轻松得出两个新的界限。首先，我们通过确定$ d = k-2 $的渐近最佳学位条件和所有$ k \ ge 3 $，以$ d = k-1 $的价格扩展了Rödl，Ruciński和Szemerédi的经典结果。 Polcyn，Reiher，Rödl和Schülke独立证明了这一点。其次，我们为紧密的汉密尔顿周期$ d $ d $ -k $均匀的超刻刻度提供了$ 1-1/（2（k-d））$的一般上限，从而将差距缩小到$ 1-1/\ sqrt {k-d} $的下限。
We develop a new framework to study minimum $d$-degree conditions in $k$-uniform hypergraphs, which guarantee the existence of a tight Hamilton cycle. Our main theoretical result deals with the typical absorption, path cover and connecting arguments for all $k$ and $d$ at once, and thus sheds light on the underlying structural problems. Building on this, we show that one can study minimum $d$-degree conditions of $k$-uniform tight Hamilton cycles by focusing on the inner structure of the neighbourhoods. This reduces the matter to an Erdős--Gallai-type question for $(k-d)$-uniform hypergraphs, which is of independent interest. Once this framework is established, we can easily derive two new bounds. Firstly, we extend a classic result of Rödl, Ruciński and Szemerédi for $d=k-1$ by determining asymptotically best possible degree conditions for $d = k-2$ and all $k \ge 3$. This was proved independently by Polcyn, Reiher, Rödl and Schülke. Secondly, we provide a general upper bound of $1-1/(2(k-d))$ for the tight Hamilton cycle $d$-degree threshold in $k$-uniform hypergraphs, thus narrowing the gap to the lower bound of $1-1/\sqrt{k-d}$ due to Han and Zhao.
