The Strong Nine Dragon Tree Conjecture is True for d ≤ 2(k+1)

The arboricity Γ(G) of an undirected graph $G =(V,E)$ is the minimal number k such that E can be partitioned into k forests on V. Nash-Williams' formula states that k = γ(G), where γ(G) is the maximum of |EH||VH|-1 over all subgraphs (VH , EH ) of G with |VH | ≥ 2. The Strong Nine Dragon Tree Conjecture states that if γ(G) ≤ k + d/d+k+1 for k, d ∈ N, then there is a partition of the edge set of G into $k + 1$ forests on V such that one forest has at most d edges in each connected component. Here we prove the Strong Nine Dragon Tree Conjecture when d ≤ 2(k +1), which is a new result for all $(k, d)$ such that $d > k + 1$. In fact, we prove a stronger theorem. We prove that a weaker sparsity notion, called $(k, d)$-sparseness, suffices to give the decomposition, under the assumption that the graph decomposes into $k+1$ forests. This is a new result for all $(k, d)$ where $d > 1$, and improves upon the recent resolution of the Overfull Nine Dragon Tree Theorem for all $(k, d)$ when d ≤ 2(k +1). As a corollary, we obtain that planar graphs of girth five decompose into a forest and a forest where every component has at most four edges, and by duality, we obtain that $5$-edge-connected planar graphs have a 4/5-thin tree, improving a result of the authors that $5$-edge-connected planar graphs have a 5/6-thin tree