A short proof of the blow-up lemma for approximate decompositions

TL;DR

This paper presents a shorter proof of the blow-up lemma for approximate decompositions, enabling almost decompositions of quasirandom graphs into bounded degree graphs with enhanced properties, impacting combinatorial design problems.

Contribution

The authors provide a significantly shorter proof of the blow-up lemma for approximate decompositions, extending its applicability to stronger quasirandom packings and combinatorial design constructions.

Findings

Shorter proof of the blow-up lemma for approximate decompositions

Enables decompositions of large complete graphs into specified bounded degree graphs

Applications to solving longstanding conjectures in graph theory and combinatorics

Abstract

Kim, K\"uhn, Osthus and Tyomkyn (Trans. Amer. Math. Soc. 371 (2019), 4655--4742) greatly extended the well-known blow-up lemma of Koml\'os, S\'ark\"ozy and Szemer\'edi 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, K\"uhn and Osthus to prove the tree packing conjecture due to Gy\'arf\'as 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, K\"uhn 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 $\varepsilon^{-1}$.