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On 1-improper 2-coloring of sparse graphs. (English) Zbl 1281.05060
Summary: A graph $$G$$ is $$(1,1)$$-colorable if its vertices can be partitioned into subsets $$V_1$$ and $$V_2$$ such that every vertex in $$G[V_i]$$ has degree at most 1 for each $$i\in\{1,2\}$$. We prove that every graph with maximum average degree at most $$\frac{14}{5}$$ is $$(1,1)$$-colorable. In particular, it follows that every planar graph with girth at least 7 is ($$1,1$$)-colorable. On the other hand, we construct graphs with maximum average degree arbitrarily close to $$\frac{14}{5}$$ (from above) that are not $$(1,1)$$-colorable.
In fact, we establish the best possible sufficient condition for the $$(1,1)$$-colorability of a graph $$G$$ in terms of the minimum, $$\rho_G$$, of $$\rho_G(S)=7| S| -5| E(G[S])|$$ over all subsets $$S$$ of $$V(G)$$. Namely, every graph $$G$$ with $$\rho_G\geq 0$$ is $$(1,1)$$-colorable. On the other hand, we construct infinitely many non-$$(1,1)$$-colorable graphs $$G$$ with $$\rho_G=-1$$. This solves a related conjecture of A. Kurek and A. Ruciński from [J. Graph Theory 18, No. 1, 73–81 (1994; Zbl 0790.05027)].

##### MSC:
 05C15 Coloring of graphs and hypergraphs 05C42 Density (toughness, etc.) 05C10 Planar graphs; geometric and topological aspects of graph theory 05C35 Extremal problems in graph theory
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