论文标题
稀疏伪随机超图中的因素和松散的汉密尔顿周期
Factors and loose Hamilton cycles in sparse pseudo-random hypergraphs
论文作者
论文摘要
我们使用以下相对较弱的伪随机概念研究了稀疏伪随机$ k $均匀的超图中跨越结构的出现。 $ n $顶点上的a $ k $ - 均匀的超graph $ h $称为$（p，α，ε）$ - pseudo-random，如果全部（不一定是脱节）顶点子集$ a_1，\ dots，a_k {\ subseteq} v（h subseteq} v（h） | a_k | {\ geQ}αn^{k} $我们有$$ e（a_1，\ dots，a_k）=（1 \pmε） $ n $）的划分假设）任何$ k $ - 均匀$ \ big（p，α，o（1）\ big）$ - pseudo-random $ n $ n $ vertex hypergraph $ h $，具有轻度的最低顶点度条件，其中包含$ f $ - f $ - f $ - factor。该方法还使我们能够在足够的伪随机超图中建立松散的汉密尔顿循环，所有结果暗示着相应的界限，即超图伪随机性的较强概念，例如jumbledness或较大的光谱间隙。 结果，$ \ big（p，α，o（o（1）\ big）$ - pseudo-random $ k $ -graphs如上所述包含：$（i）$，如果$α= o（p^{k}）$（p^{k}）$（p^{k}）$和$（II）$ a HIALILTON CAICE，如果$α= O（p^o（p^o（p^{这扩展了Lenz-Mubayi和Lenz--Mubayi-Mycroft的作品，他们研究了在密集环境中的类似问题。
We investigate the emergence of spanning structures in sparse pseudo-random $k$-uniform hypergraphs, using the following comparatively weak notion of pseudo-randomness. A $k$-uniform hypergraph $H$ on $n$ vertices is called $(p,α,ε)$-pseudo-random if for all (not necessarily disjoint) vertex subsets $A_1,\dots, A_k{\subseteq} V(H)$ with $|A_1|\cdots |A_k|{\geq}αn^{k}$ we have $$e(A_1,\dots, A_k)=(1\pmε)p |A_1|\cdots |A_k|.$$ For any linear $k$-uniform $F$ we provide a bound on $α=α(n)$ in terms of $p=p(n)$ and $F$, such that (under natural divisibility assumptions on $n$) any $k$-uniform $\big(p,α, o(1)\big)$-pseudo-random $n$-vertex hypergraph $H$ with a mild minimum vertex degree condition contains an $F$-factor. The approach also enables us to establish the existence of loose Hamilton cycles in sufficiently pseudo-random hypergraphs and all results imply corresponding bounds for stronger notions of hypergraph pseudo-randomness such as jumbledness or large spectral gap. As a consequence, $\big(p,α, o(1)\big)$-pseudo-random $k$-graphs as above contain: $(i)$ a perfect matching if $α=o(p^{k})$ and $(ii)$ a loose Hamilton cycle if $α=o(p^{k-1})$. This extends the works of Lenz--Mubayi, and Lenz--Mubayi--Mycroft who studied the analogous problems in the dense setting.
