Graph Theory Question on Planar Graphs

Question

How can it be proven that every planar simple graph is the union of three forests?

Answer 1

In addition to the references I gave in the comments, there is the paper, Roberto Grossi and Elena Lodi, Simple planar graph partition into three forests, Discrete Appl Math 84 (1998) 121-132, MR 99d:05071. According to the summary, "We describe a simple way of partitioning a planar graph into three edge-disjoint forests in $O(n\log n)$ time, where $n$ is the number of its vertices."

EDIT: I note that the converse is quite false, e.g., each of the nonplanar graphs $K_{3,3}$, $K_5$, and $K_6$ is a union of three (edge-disjoint) paths. $K_6$, for example, is the union of 123456, 241635, and 315264. The Nash-Williams paper cited in the comments gives a necessary and sufficient condition for a graph to be a union of three (or, more generally, $k$) forests.