Do planar graphs have an acyclic two-coloring?
G. Chartrand, H.V. Kronk, C.E. Wall showed in "The point-arboricity of a graph" (Israel J. Math., 6 (1968), pp. 169–175) that the vertex-set of any planar graph can be partitioned into three induced forests.
Later, Chartrand and Kronk provided an example showing that 'three' cannot be replaced by 'two', see "THE POINT-ARBORICITY OF PLANAR GRAPHS" (J. London Math. Soc., 44 (1969), pp. 612–616). It is the dual of the Tutte Graph.
I think it is still an open problem whether every planar graph with $n$ vertices has an induced forest on $n/2$ vertices.