# The Pseudoforest analogue for the Strong Nine Dragon Tree Conjecture is True

## TL;DR

This paper proves a graph decomposition result related to the Strong Nine Dragon Tree Conjecture, showing that graphs with certain average degree bounds can be decomposed into pseudoforests with controlled component sizes.

## Contribution

It establishes a pseudoforest analogue for the conjecture, providing a new decomposition theorem for graphs based on average degree constraints.

## Key findings

- Graphs with bounded average degree decompose into pseudoforests with small components.
- The result confirms a specific case of the Strong Nine Dragon Tree Conjecture.
- Provides a constructive method for such decompositions.

## Abstract

We prove that for any positive integers $k$ and $d$, if a graph $G$ has maximum average degree at most $2k + \frac{2d}{d+k+1}$, then $G$ decomposes into $k+1$ pseudoforests $C_{1},\ldots,C_{k+1}$ such that there is an $i$ such that for every connected component $C$ of $C_{i}$, we have that $e(C) \leq d$.