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

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 NashWilliams 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 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).