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

The arboricity (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 = (G), where (G) is the maximum of |E₇||V₇|-1 over all subgraphs (VH, EH) of G with |VH | 2. The Strong Nine Dragon Tree Conjecture states that if (G) k + dd+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 45-thin tree, improving a result of the authors that 5-edge-connected planar graphs have a 56-thin tree