A Proof of the Tree Packing Conjecture

TL;DR

This paper proves Gyárfás's 1976 conjecture that any family of trees with sizes from 1 to n can be packed into a complete graph of n vertices, using polynomial methods and labeling techniques.

Contribution

It introduces a novel approach by translating the tree packing problem into a labeling problem and employs polynomial methods for the proof.

Findings

Confirmed Gyárfás's conjecture for all tree families

Developed a new labeling-based proof technique

Demonstrated the effectiveness of polynomial methods in combinatorial problems

Abstract

We prove a conjecture of Gy\'arf\'as (1976), which asserts that any family of trees $T_1, \dots, T_{n}$ where each $T_k$ has $k$ vertices packs into $K_n$. We do so by translating the decomposition problem into a labeling problem, namely complete labeling. Our proof employs the polynomial method using a functional reformulation of the conjecture.