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Defective 2-colorings of sparse graphs. (English) Zbl 1282.05041
Summary: A graph $$G$$ is $$(j,k)$$-colorable if its vertices can be partitioned into subsets $$V_1$$ and $$V_2$$ such that every vertex in $$G[V_1]$$ has degree at most $$j$$ and every vertex in $$G[V_2]$$ has degree at most $$k$$. We prove that if $$k\geqslant 2j+2$$, then every graph with maximum average degree at most $$2\left(2-\frac{k+2}{(j+2)(k+1)}\right)$$ is $$(j,k)$$-colorable. On the other hand, we construct graphs with the maximum average degree arbitrarily close to $$2\left(2-\frac{k+2}{(j+2)(k+1)}\right)$$ (from above) that are not $$(j,k)$$-colorable.In fact, we prove a stronger result by establishing the best possible sufficient condition for the $$(j,k)$$-colorability of a graph $$G$$ in terms of the minimum, $$\phi_{j,k}(G)$$, of the difference $$\phi_{j,k}(W,G)=\left(2-\frac{k+2}{(j+2)(k+1)}\right)| W| -| E(G[W])|$$ over all subsets $$W$$ of $$V(G)$$. Namely, every graph $$G$$ with $$\phi_{j,k}(G)>\frac{-1}{k+1}$$ is $$(j,k)$$-colorable. On the other hand, we construct infinitely many non-$$(j,k)$$-colorable graphs $$G$$ with $$\phi_{j,k}(G)=\frac{-1}{k+1}$$.

##### MSC:
 05C15 Coloring of graphs and hypergraphs 05C42 Density (toughness, etc.) 05C35 Extremal problems in graph theory 05C07 Vertex degrees
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