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A Proof of the Tree Packing Conjecture

Parikshit Chalise, Antwan Clark, Edinah K. Gnang

Abstract

We prove a conjecture of Gyárfás (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.

A Proof of the Tree Packing Conjecture

Abstract

We prove a conjecture of Gyárfás (1976), which asserts that any family of trees where each has vertices packs into . 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.

Paper Structure

This paper contains 5 sections, 7 theorems, 78 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Proof Outline
  4. Polynomial Certificate and Composition Lemma
  5. Concluding Remarks

Key Result

Theorem 1.9

For any sequence ${\bf g} \in\left(\mathbb{Z}_{n}^{\mathbb{Z}_{n}}\right)^{n}$ of $n$ augmented functional trees, there exists a sequence $\boldsymbol{\sigma} \in (\mathrm{S}_n)^n$ of $n$ permutations such that ${\boldsymbol{\sigma} \bf g}\boldsymbol{\sigma}^{-1}$ is complete.

Figures (2)

  • Figure 1.1: $G \rightsquigarrow G_g$. Here, $g \in {\mathbb Z}_4^{{\mathbb Z}_4}$ is specified by $g(0) = 0,\; g(1) = 0,\; g(2) = 1,\text{ and } g(3) =1.$
  • Figure 1.2: A complete labeling of a sequence of augmented functional stars as outlined in Example \ref{['ex:star-seq']}. As a result, the union of the corresponding sequence of functional stars form an orientation of ${\mathbb K}_4$.

Theorems & Definitions (26)

  • Conjecture 1.1: Tree Packing Conjecture
  • Definition 1.2
  • Remark 1.3
  • Example 1.4
  • Definition 1.5: Augmented functional tree sequence
  • Definition 1.6: Complete labeling
  • Remark 1.7
  • Example 1.8
  • Theorem 1.9: Tree Packing Theorem
  • Definition 2.1
  • ...and 16 more