an:07209416 Zbl 1441.05179 Choi, Ilkyoo; Dross, Fran??ois; Ochem, Pascal Partitioning sparse graphs into an independent set and a graph with bounded size components EN Discrete Math. 343, No. 8, Article ID 111921, 16 p. (2020). 00449853 2020
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05C70 05C42 05C69 05C10 planar graphs; vertex partition; bounded component; discharging method; improper coloring; islands Summary: We study the problem of partitioning the vertex set of a given graph so that each part induces a graph with components of bounded order; we are also interested in restricting these components to be paths. In particular, we say a graph $$G$$ admits an $$( \mathcal{I} , \mathcal{O}_k )$$-partition if its vertex set can be partitioned into an independent set and a set that induces a graph with components of order at most $$k$$. We prove that every graph $$G$$ with $$\mathrm{mad}( G ) < \frac{ 5}{ 2}$$ admits an $$( \mathcal{I} , \mathcal{O}_3 )$$-partition. This implies that every planar graph with girth at least 10 can be partitioned into an independent set and a set that induces a graph whose components are paths of order at most 3. We also prove that every graph $$G$$ with $$\mathrm{mad}( G ) < \frac{ 8 k}{ 3 k + 1} = \frac{ 8}{ 3} \left( 1 - \frac{ 1}{ 3 k + 1}\right)$$ admits an $$( \mathcal{I} , \mathcal{O}_k )$$-partition. This implies that every planar graph with girth at least 9 can be partitioned into an independent set and a set that induces a graph whose components have order at most 9.