We survey recent advances in the theory of graph and hypergraph decompositions, with a focus on extremal results involving minimum degree conditions. We also collect a number of intriguing open problems, and formulate new ones.
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 Gyarfas 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 Szemeredis regularity lemma. In particular our result can be viewed as an extension of the classical blow-up lemma of Komlos, SarkH{o}zy and Szemeredi to the setting of approximate decompositions.
