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$$(k,1)$$-coloring of sparse graphs. (English) Zbl 1238.05084
Summary: A graph $$G$$ is $$(k,1)$$-colorable if the vertex set of $$G$$ can be partitioned into subsets $$V_{1}$$ and $$V_{2}$$ such that the graph $$G[V_{1}]$$ induced by the vertices of $$V_{1}$$ has maximum degree at most $$k$$ and the graph $$G[V_{2}]$$ induced by the vertices of $$V_{2}$$ has maximum degree at most 1. We prove that every graph with maximum average degree less than $$\frac {10k+22}{3k+9}$$ admits a $$(k,1)$$-coloring, where $$k\geq 2$$. In particular, every planar graph with girth at least 7 is (2,1)-colorable, while every planar graph with girth at least 6 is (5,1)-colorable. On the other hand, when $$k\geq 2$$ we construct non-$$(k,1)$$-colorable graphs whose maximum average degree is arbitrarily close to $$\frac {14k}{4k+1}$$.

##### MSC:
 05C15 Coloring of graphs and hypergraphs 05C42 Density (toughness, etc.) 05C10 Planar graphs; geometric and topological aspects of graph theory
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