An Approximate Version of the Strong Nine Dragon Tree Conjecture

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