# Edge Disjoint Caterpillar Realizations

## TL;DR

This paper investigates conditions for the existence of edge disjoint caterpillar realizations of tree degree sequences, providing necessary and sufficient criteria and proving a conjecture for specific cases.

## Contribution

It establishes necessary and sufficient conditions for edge disjoint caterpillar realizations and proves the conjecture for cases with up to four sequences or sufficiently large vertex counts.

## Key findings

- Characterization of when two tree degree sequences have edge disjoint caterpillar realizations.
- Proof of the conjecture for up to four sequences.
- Proof of the conjecture for large n relative to k.

## Abstract

In this paper, we consider the edge disjoint caterpillar realizations of tree degree sequences. We give the necessary and sufficient conditions when two tree degree sequences have edge disjoint caterpillar realizations. We conjecture that an arbitrary number of tree degree sequences have edge disjoint realizations if every vertex is a leaf in at most one tree. We prove that the conjecture is true if the number of tree degree sequences is at most $4$. We also prove that the conjecture is true if $n \ge \max\{22k-11, 396\}$, where $n$ is the number of vertices and $k$ is the number of tree degree sequences.