Decomposition of sparse graphs into forests: The Nine Dragon Tree Conjecture for k ≤ 2

For a loopless multigraph G, the fractional arboricityArb(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+dk+d+1, then G decomposes into k+1 forests with one having maximum degree at most d. The conjecture was previously proved for d=k+1 and for k=1 when d≤6. We prove it for all d when k≤2, except for (k,d)=(2,1).