Covering planar graphs with forests, one having bounded maximum degree

We prove that every planar graph has an edge partition into three forests, one having maximum degree at most 4. This answers a conjecture of Balogh, Kochol, Pluhár and Yu [J. Balogh, M. Kochol, A. Pluhár, X. Yu, Covering planar graphs with forests, J. Combin. Theory Ser. B. 94 (2005) 147–158]. We also prove that every planar graph with girth g⩾6 (resp. g⩾7) has an edge partition into two forests, one having maximum degree at most 4 (resp. 2).