AI 中文总结
本文证明了随机图生成子图在最小度条件下可包含所有满足特定三角形外顶点条件的有界度图，给出了Sauer-Spencer定理的稀疏局部韧性版本，并允许H为扩张图。
AI 中文摘要
我们证明，对于所有 $\Delta \geq 2$ 和 $\gamma > 0$，存在常数 $C = C(\Delta, \gamma)$，使得当 $p\geq C(\log n/n)^{1/\Delta}$ 时，渐近几乎必然地，$G(n,p)$ 的每个最小度至少为 $(1-1/(2\Delta)+\gamma)pn$ 的生成子图 $G$ 都包含每个最大度至多为 $\Delta$ 且至少有 $Cp^{-2}$ 个顶点不在 $H$ 的任何三角形中的 $n$ 顶点图 $H$。这是 Sauer 和 Spencer 经典定理的“稀疏局部韧性版本”。$H$ 应包含一些不在三角形中的顶点的条件是必要的，事实上，$p^{-2}$ 这个量是渐近最优的。我们结果的一个关键特征是允许 $H$ 是扩张图，这使其区别于先前类似性质的结果，那些结果处理的是例如次线性带宽的图。我们的证明利用了正则性论证，其中随机图的稀疏爆破引理是关键工具。
英文摘要
We prove that for all $Δ\geq 2$ and $γ> 0$, there exists a constant $C = C(Δ, γ)$ such that for $p\geq C(\log n/n)^{1/Δ}$, asymptotically almost surely, every spanning subgraph $G$ of $G(n,p)$ with minimum degree at least $(1-1/(2Δ)+γ)pn$ contains every $n$-vertex graph $H$ with maximum degree at most $Δ$ and with at least $Cp^{-2}$ vertices not in any triangles of $H$. This is a 'sparse local resilience version' of a classical theorem of Sauer and Spencer. The condition that $H$ should contain some vertices not in triangles is necessary, and in fact, the quantity $p^{-2}$ is asymptotically best possible. A key feature of our result is that $H$ is allowed to be an expander graph, distinguishing it from previous results of similar nature, which dealt with, e.g., graphs of sublinear bandwidth. Our proof makes use of regularity arguments, with the sparse blow-up lemma for random graphs being a key tool.
Comments22 pages, no figures
