Equitable partition of graphs into induced forests

An equitable partition of a graph G is a partition of the vertex-set of G such that the sizes of any two parts differ by at most one. We show that every graph with an acyclic coloring with at most k colors can be equitably partitioned into k - 1 induced forests. We also prove that for any integers d ? 1 and k ? 3 d - 1 , any d -degenerate graph can be equitably partitioned into k induced forests.Each of these results implies the existence of a constant c such that for any k ? c , any planar graph has an equitable partition into k induced forests. This was conjectured by Wu, Zhang, and Li in 2013.