Near-proper vertex 2-colorings of sparse graphs

A graph $G$ is $(2,1)$-colorable if its vertices can be partitioned into subsets $V_1$ and $V_2$ such that in $G[V_1]$ any component contains at most two vertices while $G[V_2]$ is edgeless. We prove that every graph $G$ with the maximum average degree $\mathrm{mad}(G)$ smaller than 7/3 is $(2,1)$-colorable. It follows that every planar graph with girth at least 14 is $(2,1)$-colorable. We also construct a planar graph $G_n$ with $\mathrm{mad}(G_n)=(18n-2)/(7n-1)$ that is not $(2,1)$-colorable. Bibl. 5.