Partition of Sparse Graphs into Two Forests with Bounded Degree

Borodin and Kostochka proved that for d₂ 2d₁+2 and a graph G where every subgraph H satisfies e (H) < (2 - d₂+2 (d₁+2) (d₂+1) ) n (H) + 1d₂+1 has a vertex partition V (G) = V₁ V₂ such that GVᵢ has maximum degree at most dᵢ for each i. We show that under the same conditions we can additionally conclude that each GVᵢ is a forest.