A blow-up lemma for approximate decompositions

TL;DR

This paper introduces a new method for approximate graph decompositions, extending the blow-up lemma to dense graphs, enabling solutions to longstanding problems like tree packings and the Oberwolfach problem.

Contribution

It develops a novel approach for approximate decompositions of dense graphs into sparse graphs, extending the classical blow-up lemma to this setting.

Findings

Achieved approximate decompositions for dense graphs into bounded degree subgraphs.

Extended the blow-up lemma to approximate decompositions, applicable to super-regular graphs.

Provided solutions to the tree packing conjecture and Oberwolfach problem in an approximate form.

Abstract

We develop a new method for constructing approximate decompositions of dense graphs into sparse graphs and apply it to longstanding decomposition problems. For instance, our results imply the following. Let $G$ be a quasi-random $n$-vertex graph and suppose $H_1,\dots,H_s$ are bounded degree $n$-vertex graphs with $\sum_{i=1}^{s} e(H_i) \leq (1-o(1)) e(G)$. Then $H_1,\dots,H_s$ can be packed edge-disjointly into $G$. The case when $G$ is the complete graph $K_n$ implies an approximate version of the tree packing conjecture of Gy\'arf\'as and Lehel for bounded degree trees, and of the Oberwolfach problem. We provide a more general version of the above approximate decomposition result which can be applied to super-regular graphs and thus can be combined with Szemer\'edi's regularity lemma. In particular our result can be viewed as an extension of the classical blow-up lemma of Koml\'os, S\'ark\H{o}zy and Szemer\'edi to the setting of approximate decompositions.