Abstract
A graph $G$ is $(k,0)$-colorable if its vertices can be partitioned into subsets $V_1$ and $V_2$ such that in $G[V_1]$ every vertex has degree at most $k$, while $G[V_2]$ is edgeless. For every integer $k\ge 1$, we prove that every graph with the maximum average degree smaller than $\frac {3k+4}{k+2}$ is $(k,0)$-colorable. In particular, it follows that every planar graph with girth at least $7$ is $(8,0)$-colorable. On the other hand, we construct planar graphs with girth $6$ that are not $(k,0)$-colorable for arbitrarily large $k$.