The sparse Strong Nine Dragon Tree Conjecture

Let $k$ and $d$ be positive integers. For a graph $G$, call it $(k,d)$-sparse if every subgraph $H$ satisfies

$$(k+1)(k+d)v(H)-(k+d+1)e(H)-k^2\geq 0.$$

Call a subgraph $H$ $(k+1)$-overfull if $e(H)>(k+1)(v(H)-1)$. Sparse Strong Nine Dragon Tree Conjecture. Every $(k,d)$-sparse graph with no $(k+1)$-overfull subgraph decomposes into $k+1$ forests such that one forest has every component containing at most $d$ edges.

This conjecture strengthens the paper’s main theorem, which proves the assertion when $d\leq 2(k+1)$. It is motivated by the fact that $(k,d)$-sparsity is a relaxed form of the fractional-arboricity inequality, while excluding overfull subgraphs ensures decomposition into $k+1$ forests.