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
j
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.