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

The vertex arboricity $va\left(G\right)$ 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 $va\left(G\right)\le 3$ for every planar graph $G$. In this paper, we prove that $va\left(G\right)\le 2$ 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 $k\in \{3,4,5,6\}$, planar graphs $G$ without $k$-cycles have $va\left(G\right)\le 2$.