Covering planar graphs with forests, one having a 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 et al. (J. Combin. Theory 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 4 (resp. 2).