Optimal packings of bounded degree trees

Abstract

We prove that if T-1, ..., T-n is a sequence of bounded degree trees such that T-i has i vertices, then K-n has a decomposition into T-1, ..., T-n. This shows that the tree packing conjecture of Gy ' arf ' as 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 Szemeredi'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.