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Improper coloring of sparse graphs with a given girth. I: $$(0,1)$$-colorings of triangle-free graphs. (English) Zbl 1297.05083
Summary: A graph $$G$$ is $$(0, 1)$$-colorable if $$V(G)$$ can be partitioned into two sets $$V_0$$ and $$V_1$$ so that $$G [V_0]$$ is an independent set and $$G [V_1]$$ has maximum degree at most 1. The problem of verifying whether a graph is $$(0, 1)$$-colorable is NP-complete even in the class of planar graphs of girth 9. The maximum average degree, $$\operatorname{mad}(G)$$, of a graph $$G$$ is the maximum of $$\frac{2 | E(H) |}{| V(H) |}$$ over all subgraphs $$H$$ of $$G$$. It was proved recently that every graph $$G$$ with $$\operatorname{mad}(G) \leq \frac{12}{5}$$ is $$(0, 1)$$-colorable, and this is sharp. This yields that every planar graph with girth at least 12 is $$(0, 1)$$-colorable. Let $$F(g)$$ denote the supremum of $$a$$ such that for some constant $$b_g$$ every graph $$G$$ with girth $$g$$ and $$| E(H) | \leq a | V(H) | + b_g$$ for every $$H \subseteq G$$ is $$(0, 1)$$-colorable. By the above, $$F(3) = 1.2$$.
We find the exact value for $$F(4)$$ and $$F(5)$$: $$F(4) = F(5) = \frac{11}{9}$$. In fact, we also find the best possible values of $$b_4$$ and $$b_5$$: every triangle-free graph $$G$$ with $$| E(H) | < \frac{11 | V(H) | + 5}{9}$$ for every $$H \subseteq G$$ is $$(0, 1)$$-colorable, and there are infinitely many not $$(0, 1)$$-colorable graphs $$G$$ with girth 5, $$| E(G) | = \frac{11 | V(G) | + 5}{9}$$ and $$| E(H) | < \frac{11 | V(H) | + 5}{9}$$ for every proper subgraph $$H$$ of $$G$$. A corollary of our result is that every planar graph of girth 11 is $$(0, 1)$$-colorable. This answers a half of a question by P. Dorbec et al. [J. Graph Theory 75, No. 2, 191–202 (2014; Zbl 1280.05040)]. In a companion paper, we show that for every $$g$$, $$F(g) \leq 1.25$$ and resolve some similar problems for the so-called $$(j, k)$$-colorings generalizing $$(0, 1)$$-colorings.

##### 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
##### Keywords:
colorable graphs; $$(j, k)$$-colorings
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##### References:
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