On the tree packing conjecture

TL;DR

This paper advances the understanding of the Gyárfás tree packing conjecture by establishing partial packing results for sets of trees with specific size and degree constraints into complete graphs.

Contribution

It proves new bounds on the number of trees that can be packed into complete graphs under various conditions, extending previous conjectures and using advanced embedding theorems.

Findings

Packed 1/10 n^{1/4} trees into K_{n+1} for large n.

Packed 1/10 n^{1/4} trees with no stars into K_{n}.

Packed 1/4 n^{1/3} trees with high maximum degree into K_n.

Abstract

The Gy\'arf\'as tree packing conjecture states that any set of $n-1$ trees $T_{1},T_{2},..., T_{n-1}$ such that $T_i$ has $n-i+1$ vertices pack into $K_n$. We show that $t=1/10n^{1/4}$ trees $T_1,T_2,..., T_t$ such that $T_i$ has $n-i+1$ vertices pack into $K_{n+1}$ (for $n$ large enough). We also prove that any set of $t=1/10n^{1/4}$ trees $T_1,T_2,..., T_t$ such that no tree is a star and $T_i$ has $n-i+1$ vertices pack into $K_{n}$ (for $n$ large enough). Finally, we prove that $t=1/4n^{1/3}$ trees $T_1,T_2,..., T_t$ such that $T_i$ has $n-i+1$ vertices pack into $K_n$ as long as each tree has maximum degree at least $2n^{2/3}$ (for $n$ large enough). One of the main tools used in the paper is the famous spanning tree embedding theorem of Koml\'os, S\'ark\"ozy and Szemer\'edi.