On the vertex-arboricity of planar graphs

The vertex-arboricity a ( G ) of a graph G is the minimum number of subsets into which the set of vertices of G can be partitioned so that each subset induces a forest. It is well-known that a ( G ) ≤ 3 for any planar graph G . In this paper we prove that a ( G ) ≤ 2 whenever G is planar and either G has no 4-cycles or any two triangles of G are at distance at least 3.