Beyond the Pseudoforest Strong Nine Dragon Tree Theorem

TL;DR

This paper strengthens the Strong Nine Dragon Tree Conjecture by providing a decomposition with additional properties such as acyclicity and bounded diameter of components, which are proven to be optimal.

Contribution

It introduces a refined decomposition theorem with diameter bounds for components, extending previous results and proving their optimality.

Findings

Decomposition where the pseudoforest component is acyclic with bounded diameter.

Diameter bounds are proven to be best possible extensions.

Results hold even when only maximum degree constraints are enforced.

Abstract

The pseudoforest version of the Strong Nine Dragon Tree Conjecture states that if a graph $G$ has maximum average degree $\text{mad}(G) = 2 \max_{H \subseteq G} \frac{e(G)}{v(G)}$ at most $2(k + \frac{d}{k+d+1})$, then it has a decomposition into $k+1$ pseudoforests where in one pseudoforest $F$ the components of $F$ have at most $d$ edges. This was proven in 2020. We strengthen this theorem by showing that we can find such a decomposition where additionally $F$ is acyclic, the diameter of the components of $F$ is at most $2\ell + 2$, where $\ell = \lfloor\frac{d-1}{k+1} \rfloor$, and at most $2\ell + 1$ if $d \equiv 1 \bmod k+1$. Furthermore, for any component $K$ of $F$ and any $z \in \mathbb N$, we have $diam(K) \leq 2z$ if $e(K) \geq d - z(k-1) + 1$. We also show that both diameter bounds are best possible as an extension for both the Strong Nine Dragon Tree Conjecture for pseudoforests and its original conjecture for forests. In fact, they are still optimal even if we only enforce $F$ to have any constant maximum degree, instead of enforcing every component of $F$ to have at most $d$ edges.