On the vertex-arboricity of planar graphs without 7-cycles

The vertex arboricity v a ( G ) of a graph G is the minimum number of colors the vertices can be labeled so that each color class induces a forest. It was well-known that v a ( G ) ≤ 3 for every planar graph G . In this paper, we prove that v a ( G ) ≤ 2 if G is a planar graph without 7-cycles. This extends a result in [A. Raspaud, W. Wang, On the vertex-arboricity of planar graphs, European J. Combin. 29 (2008) 1064–1075] that for each k ∈ { 3 , 4 , 5 , 6 } , planar graphs G without k -cycles have v a ( G ) ≤ 2 .