论文标题
随机的汉密尔顿周期宽松$ 3 $ - 均匀的超图的转移
Transference for loose Hamilton cycles in random $3$-uniform hypergraphs
论文作者
论文摘要
超图中的汉密尔顿周期松散的是覆盖所有顶点的边缘循环序列，其中只有两个连续的边缘相交，并在一个顶点中恰好在一个顶点中进行操作。考虑到Dirac的定理，自然要问哪种最低$ D $数量的条件可以保证在$ K $均匀的超盖中存在松散的汉密尔顿周期。对于$ k = 3 $，每个$ d \ in \ {1,2 \} $中的每个$ d \，必要和充分的条件已知。我们表明，这些结果遵循“转移原理”对它们稀疏的随机类似物。证明结合了图设置中的几个想法，并依赖于吸收方法。特别是，我们采用了夸恩和费伯的一种新型方法，通过收缩程序在稀疏超图的亚图中找到吸收者。在$ d = 2 $的情况下，我们的发现在渐近上是最佳的。
A loose Hamilton cycle in a hypergraph is a cyclic sequence of edges covering all vertices in which only every two consecutive edges intersect and do so in exactly one vertex. With Dirac's theorem in mind, it is natural to ask what minimum $d$-degree condition guarantees the existence of a loose Hamilton cycle in a $k$-uniform hypergraph. For $k=3$ and each $d \in \{1,2\}$, the necessary and sufficient such condition is known precisely. We show that these results adhere to a `transference principle' to their sparse random analogues. The proof combines several ideas from the graph setting and relies on the absorbing method. In particular, we employ a novel approach of Kwan and Ferber for finding absorbers in subgraphs of sparse hypergraphs via a contraction procedure. In the case of $d = 2$, our findings are asymptotically optimal.
