An Approximate Version of the Strong Nine Dragon Tree Conjecture

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 $d leq k+1$. 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 (2lceil frac{d}{k+1} +2 rceil)^{lceil frac{d}{k+1} + 2) rceil} - k)}{k+1} $ edges.