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. (C) 2015 Elsevier B.V. All rights reserved.