Strong Nine Dragon Tree Conjecture

Let $G$ be a graph, and let $d$ and $k$ be positive integers. Define the fractional arboricity of $G$ by

$$\gamma(G)=\max_{H\subseteq G,\,v(H)\geq 2}\frac{e(H)}{v(H)-1}.$$

Strong Nine Dragon Tree Conjecture. If

$$\gamma(G)\leq k+\frac{d}{d+k+1},$$

then there is a partition of $G$ into $k+1$ forests such that, in one forest, every connected component has at most $d$ edges.

The conjecture strengthens the Nine Dragon Tree Theorem, which requires only that one forest have maximum degree $d$. It is known when $d\leq k+1$ and when $d\leq 2(k+1)$; all other cases remain open.