Optimal packings of bounded degree trees

TL;DR

This paper proves that dense graphs can be decomposed into bounded degree trees, confirming long-standing conjectures for this class of trees using advanced combinatorial techniques.

Contribution

It establishes the tree packing and Ringel's conjectures for all bounded degree trees by developing a general decomposition theorem for dense quasi-random graphs.

Findings

Decomposition of dense graphs into bounded degree trees confirmed

Tree packing conjecture holds for all bounded degree trees

Ringel's conjecture holds for all bounded degree trees

Abstract

We prove that if $T_1,\dots, T_n$ is a sequence of bounded degree trees so that $T_i$ has $i$ vertices, then $K_n$ has a decomposition into $T_1,\dots, T_n$. 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 Szemer\'{e}di'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.