An approximate version of the Tree Packing Conjecture

Abstrakt

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 $(2^n)$ edges in total packs into $K_{(1+\epsilon)n} . This implies asymptotic versions of the Tree Packing Conjecture of Gyárfás 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.