# Packing tree degree sequences

## TL;DR

This paper investigates the problem of packing multiple tree degree sequences into edge-disjoint trees, proving the conjecture for up to 4 sequences, with computational support for 5, and establishing conditions for arbitrary k.

## Contribution

It provides new proofs for the packing conjecture for 4 and 5 tree degree sequences, and general conditions for any number of sequences without common leaves.

## Key findings

- Proof of the conjecture for 4 tree degree sequences.
- Computer-aided proof for 5 tree degree sequences.
- General conditions for k sequences with non-leaf vertices.

## Abstract

We consider packing tree degree sequences in this paper. We set up a conjecture that any arbitrary number of tree degree sequences without common leaves have edge disjoint tree realizations. This conjecture is known to be true for $2$ and $3$ tree degree sequences. In this paper, we give a proof for $4$ tree degree sequences and a computer aided proof for $5$ tree degree sequences. We also prove that for arbitrary $k$, $k$ tree degree sequences without common leaves and at least $2k-4$ vertices which are not leaves in any of the trees always have edge disjoint tree realizations. The main ingredient in all of the presented proofs is to find rainbow matchings in certain configurations.