AI 中文总结
该研究针对稠密图的随机导出子图，证明了满足最小度条件时F-因子存在的概率下界，该下界渐近最优，还将结果推广到超图完美匹配，证明结合了多种数学工具。
AI 中文摘要
设F是一个含r个顶点的图，本文研究稠密图的随机导出子图中的F-因子问题。我们证明：对任意含r个顶点的图F和任意γ>0，若H是含n个顶点且最小度至少为(1−1/χ_cr(F)+γ)n的图，则对每个固定的p∈(0,1)，随机导出子图H[p]包含F-因子的概率至少为1/(rq)−oₙ(1)，其中q∈ℕ是由H定义的某个陪集群的阶。该概率对无穷多的F和H而言是渐近最优的，且得出H的子集有1/(rq)−oₙ(1)的比例导出F-因子，有趣的是，无论H自身是否允许F-因子，该结论均成立。在最小度条件下，超图的完美匹配也得到了类似结果。我们的证明结合了集中不等式、ℤᵈ中的格点计数以及稠密（超）图中F-因子的结构定理。
英文摘要
Let $F$ be an $r$-vertex graph. In this paper, we study the $F$-factor problem in random induced subgraphs of dense graphs. We show that for any $r$-vertex graph $F$ and $γ>0$, if $H$ is an $n$-vertex graph with minimum degree at least $(1-1/χ_{cr}(F)+γ)n$, then for every fixed $p \in (0,1)$, the random induced subgraph $H[p]$ contains an $F$-factor with probability at least $1/(rq)-o_n(1)$, where $q\in \mathbb{N}$ is the order of certain coset group defined from $H$. The probability is asymptotically best possible for infinitely many $F$ and $H$ and yields that a $1/(rq)-o_n(1)$ proportion of the subsets of $H$ induce $F$-factors, interestingly, regardless of whether $H$ itself admits an $F$-factor. Similar results are obtained for perfect matchings in hypergraphs under minimum degree conditions. Our proof combines concentration inequalities, lattice point counting in $\mathbb{Z}^d$ and structural theorems for $F$-factors in dense (hyper)graphs.
Comments20 pages
