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

The vertex arboricity va(G)va(G) of a graph GG is the minimum number of colors the vertices can be labeled so that each color class induces a forest. It was well-known that va(G)≤3va(G)≤3 for every planar graph GG. In this paper, we prove that va(G)≤2va(G)≤2 if GG 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}k∈{3,4,5,6}, planar graphs GG without kk-cycles have va(G)≤2va(G)≤2.