Decomposition of sparse graphs into forests and a graph with bounded degree

For a loopless multigraph G, the fractional arboricity Arb(G) is the maximum of |E(H)|/|V(H)|-1 over all subgraphs H with at least two vertices. Generalizing the Nash-Williams Arboricity Theorem, the Nine Dragon Tree Conjecture asserts that if Arb (G)≤ k + d/k+d+1, then G decomposes into k+1 forests with one having maximum degree at most d. The conjecture was previously proved for (k,d)∈ {(1,1),(1,2)}; we prove it for d=k+1 and when k=1 and d≤6. For (k,d)=(1,2), we can further restrict one forest to have at most two edges in each component. For general (k,d), we prove weaker conclusions. If d>k, then Arb (G)≤k + d/k+d+1 implies that G decomposes into k forests plus a multigraph (not necessarily a forest) with maximum degree at most d. If d≤k, then Arb (G)≤k + d/2k+2 implies that G decomposes into k+1 forests, one having maximum degree at most d. Our results generalize earlier results about decomposition of sparse planar graphs.