Do planar graphs have an acyclic two-coloring?

A graph has an acyclic two-coloring if its vertices can be colored with two colors such that each color class spans a forest.

Does every planar graph have an acyclic two-coloring?

An affirmative answer would imply the four-color theorem, so I guess the answer has to be no, but I've failed to find a counterexample. Or would this problem be equivalent to the four-color theorem?

• The edges of any planar graph can be decomposed into $3$ forests, and this is tight ( en.m.wikipedia.org/wiki/Arboricity ), but you aren't decomposing all the edges, and your forests are induced subgraphs. Hmm... – Pat Devlin Nov 23 '16 at 13:06

I think it is still an open problem whether every planar graph with $n$ vertices has an induced forest on $n/2$ vertices.