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Vertex decompositions of sparse graphs into an edgeless subgraph and a subgraph of maximum degree at most $$k$$. (English) Zbl 1209.05177
Summary: 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 $$k \geq 0$$, we prove that every graph with the maximum average degree smaller than $$(3k+4)/(k+2)$$ 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 $$k$$.

##### MSC:
 05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.) 05C15 Coloring of graphs and hypergraphs
##### Keywords:
vertex decompositions; sparse graphs
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##### References:
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