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

The vertex arboricity nu 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 nu a(G) <= 3 for every planar graph G. In this paper, we prove that nu 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 is an element of {3, 4, 5, 6}, planar graphs G without k-cycles have va(G) <= 2. (C) 2012 Elsevier B.V. All rights reserved.