An Approximate Version of the Strong Nine Dragon Tree Conjecture

Abstract

The Strong Nine Dragon Tree Conjecture asserts that for any integers $k$ and $d$ any graph with fractional arboricity at most $k+\frac{d}{d+k+1}$ decomposes into $k+1$ forests, such that for at least one of the forests, every connected component contains at most $d$ edges. We prove this conjecture when $d\le k+1$. We also prove an approximate version of this conjecture, that is, we prove that for any positive integers $k$ and $d$, any graph with fractional arboricity at most $k+\frac{d}{d+k+1}$ decomposes into $k+1$ forests, such that one for at least one of the forests, every connected component contains at most $d+\frac{d(k(2\lceil \frac{d}{k+1}+2\rceil {)}^{\lceil \frac{d}{k+1}+2)\rceil }-k)}{k+1}$ edges.

20 pages. The proof of Lemma 5.6 is inaccurate, as the legal order may change. As such, the proof fails. A recovery of this lemma (or something similar) would fix the proof. I am attempting to fix the error, but until then one should assume the result is incorrect