An Approximate Version of the Strong Nine Dragon Tree Conjecture

TL;DR

This paper proves an approximate version of the Strong Nine Dragon Tree Conjecture, establishing conditions under which a graph's edges can be partitioned into forests with bounded component sizes based on a density parameter.

Contribution

The authors provide a proof for an approximate form of the conjecture, replacing the original edge bound with a more flexible inequality involving graph density and component size constraints.

Findings

Confirmed the conjecture under a relaxed edge bound condition.

Established a partitioning method for graph edges into forests with size constraints.

Linked graph density to feasible forest decompositions.

Abstract

We prove the Strong Nine Dragon Tree Conjecture is true if we replace the edge bound with $d + \big\lceil k \big\lfloor\frac{d-1}{k+1}\big\rfloor \big(\frac{d}{k+1} - \frac{1}{2} \big\lceil\frac{d}{k+1}\big\rceil \big)\big\rceil \leq d + \frac{k}{2} \cdot \big(\frac{d}{k+1}\big)^2$. More precisely: let $G$ be a graph, let $d$ and $k$ be positive integers and $\gamma(G) = \max_{H \subseteq G, v(H) \geq 2} \frac{e(H)}{v(H) - 1}$. If $\gamma(G) \leq k + \frac{d}{d + k + 1}$, then there is a partition of $E(G)$ into $k + 1$ forests, where in one forest every connected component has at most $d + \big\lceil k \big\lfloor\frac{d-1}{k+1}\big\rfloor \big(\frac{d}{k+1} - \frac{1}{2} \big\lceil\frac{d}{k+1}\big\rceil \big)\big\rceil$ edges.