Vertex decompositions of sparse graphs into an independent vertex set and a subgraph of maximum degree at most 1

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A graph G is (1, 0)-colorable if its vertex set can be partitioned into subsets V1 and V0 so that in G[V1] every vertex has degree at most 1, while G[V0] is edgeless. We prove that every graph with maximum average degree at most \(\tfrac{{12}} {5} \) is (1, 0)-colorable. In particular, every planar graph with girth at least 12 is (1, 0)-colorable. On the other hand, we construct graphs with the maximum average degree arbitrarily close (from above) to \(\tfrac{{12}} {5} \) which are not (1, 0)-colorable.