论文标题
大约分解的爆炸引理的简短证明
A short proof of the blow-up lemma for approximate decompositions
论文作者
论文摘要
Kim，Kühn，Osthus和Tyomkyn（Trans。Amer。Math。Soc。371（2019），4655---4742）极大地扩展了Komlós，Sárközy和Szemerédi的众所周知的爆炸震颤，几乎可以通过任何decomptitients dencompite decomptite decompite decompite decomptite decomptient，yourcompient of decomptient yourpers and commocts yourpers commocts y mulders commocts yourcland suprands y mulderm squasts y mulderm squasts''收集具有相同多部分结构和较少边缘的有界度图。 Joos，Kim，Kühn和Osthus已经使用了这一结果，以证明由于Gyárfás和1976年的Lehel引起的树木包装猜想，以及1963年的Ringel从1963年开始对有限程度的树木的猜想，以及在Oberwolfach问题的最新解决方案中（1967年由Glock in Glock，Kim，Kim，Kim，Kim，Kim，kuth和ostos）。在这里，我们提出了一个新的且明显的证明，证明了大约分解的爆炸引理。实际上，我们证明了一个更一般的定理，该定理产生具有更强的quasirandom属性的包装，因此可以将其与Keevash在设计上的结果结合使用，以获得以下形式的结果。对于所有$ \ varepsilon> 0 $，$ r \ in \ mathbb {n} $和所有大$ n $（这样$ r $ $ r $ divide $ n-1 $），$ k_n $分解为任何$ r $ r $ r $ r $ regratular Graphs $ h_1，\ ldots，\ ldots，\ ldots，\ ldots，h_______ {n-1）$ $ h_1，\ ldots，h _ {\ varepsilon n} $每个至少包含$ \ varepsilon n $顶点，其中最多是$ \ varepsilon^{ - 1} $。
Kim, Kühn, Osthus and Tyomkyn (Trans. Amer. Math. Soc. 371 (2019), 4655--4742) greatly extended the well-known blow-up lemma of Komlós, Sárközy and Szemerédi by proving a `blow-up lemma for approximate decompositions' which states that multipartite quasirandom graphs can be almost decomposed into any collection of bounded degree graphs with the same multipartite structure and slightly fewer edges. This result has already been used by Joos, Kim, Kühn and Osthus to prove the tree packing conjecture due to Gyárfás and Lehel from 1976 and Ringel's conjecture from 1963 for bounded degree trees as well as implicitly in the recent resolution of the Oberwolfach problem (asked by Ringel in 1967) by Glock, Joos, Kim, Kühn and Osthus. Here we present a new and significantly shorter proof of the blow-up lemma for approximate decompositions. In fact, we prove a more general theorem that yields packings with stronger quasirandom properties so that it can be combined with Keevash's results on designs to obtain results of the following form. For all $\varepsilon>0$, $r\in \mathbb{N}$ and all large $n$ (such that $r$ divides $n-1$), there is a decomposition of $K_n$ into any collection of $r$-regular graphs $H_1,\ldots,H_{(n-1)/r}$ on $n$ vertices provided that $H_1,\ldots,H_{\varepsilon n}$ contain each at least $\varepsilon n$ vertices in components of size at most $\varepsilon^{-1}$.
