An approximate version of the Tree Packing Conjecture

Abstract

We prove that for any pair of constants $\epsilon >0$ and $\Delta$ and for $n$ sufficiently large, every family of trees of orders at most $n$, maximum degrees at most $\Delta$, and with at most $\left(\genfrac{}{}{0ex}{}{n}{2}\right)$ edges in total packs into $K_{(1+\epsilon )n}$. This implies asymptotic versions of the Tree Packing Conjecture of Gyarfas from 1976 and a tree packing conjecture of Ringel from 1963 for trees with bounded maximum degree. A novel random tree embedding process combined with the nibble method forms the core of the proof.

38 pages, 2 figures; suggestions by an anonymous referee incorporated; accepted to Israel J Math

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• Tree decompositions of graphs without large bipartite holes
• The tree packing conjecture for trees of almost linear maximum degree