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 H1, . . . , Hs are bounded degree n-vertex graphs with s i=1 e(Hi) ≤ (1 − o(1))e(G). Then H1, . . . , Hs can be packed edge-disjointly into G. The case when G is the complete graph Kn implies an approximate version of the tree packing conjecture of Gyárfás 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 Szemerédi's regularity lemma. In particular our result can be viewed as an extension of the classical blow-up lemma of Komlós, Sárkőzy and Szemerédi to the setting of approximate decompositions.

We prove that if T1, . . . , Tn is a sequence of bounded degree trees so that Ti has i vertices, then Kn has a decomposition into T1, . . . , Tn. This shows that the tree packing conjecture of Gyárfás and Lehel from 1976 holds for all bounded degree trees (in fact, we can allow the first o(n) trees to have arbitrary degrees). Similarly, we show that Ringel's conjecture from 1963 holds for all bounded degree trees. We deduce these results from a more general theorem, which yields decompositions of dense quasi-random graphs into suitable families of bounded degree graphs. Our proofs involve Szemerédi's regularity lemma, results on Hamilton decompositions of robust expanders, random walks, iterative absorption as well as a recent blow-up lemma for approximate decompositions.

Hadwiger's conjecture asserts that if a simple graph G has no K t+1 minor, then its vertex set V (G) can be partitioned into t stable sets. This is still open, but we prove under the same hypotheses that V (G) can be partitioned into t sets X 1 , . . . , X t , such that for 1 ≤ i ≤ t, the subgraph induced on X i has maximum degree at most a function of t. This is sharp, in that the conclusion becomes false if we ask for a partition into t − 1 sets with the same property.