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Improper coloring of sparse graphs with a given girth. II: Constructions. (English) Zbl 1333.05115
Summary: A graph $$G$$ is $$(j,k)$$-colorable if $$V(G)$$ can be partitioned into two sets $$V_j$$ and $$V_k$$ so that the maximum degree of $$G[V_j]$$ is at most $$j$$ and of $$G[V_k]$$ is at most $$k$$. While the problem of verifying whether a graph is $$(0, 0)$$-colorable is easy, the similar problem with $$(j,k)$$ in place of $$(0, 0)$$ is NP-complete for all nonnegative $$j$$ and $$k$$ with $$j+k\geq1$$. Let $$F_{j,k}(g)$$ denote the supremum of all $$x$$ such that for some constant $$c_g$$ every graph $$G$$ with girth $$g$$ and $$|E(H)|+c_g$$ for every $$H\subseteq G$$ is $$(j,k)$$-colorable. It was proved recently that $$F_{0,1}(3) =1.2$$. In a companion paper, we find the exact value $$F{0,1}(4)=F_{0,1}(5)=\frac{11}{9}$$. In this article, we show that increasing $$g$$ from $$5$$ further on does not increase $$F_{0,1}(g)\leq1.25$$ much. Our constructions show that for every $$g$$, $$F_{0,1}(g)\leq1.25$$. We also find exact values of $$F_{j,k}(g)$$ for all $$g$$ and all $$k\geq2j+2$$.
For Part I see [the authors, Eur. J. Comb. 42, 26.-48 (2014; Zbl 1297.05083)].

##### MSC:
 05C15 Coloring of graphs and hypergraphs 05C42 Density (toughness, etc.)
##### Keywords:
improper coloring; defective coloring; sparse graph; girth
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