Colouring planar graphs with bounded monochromatic components

Borodin and Ivanova proved that every planar graph of girth at least 7 is 2-choosable with the property that each monochromatic component is a path with at most 3 vertices. Axenovich et al. proved that every planar graph of girth 6 is 2-choosable so that each monochromatic component is a path with at most 15 vertices. We improve both these results by showing that planar graphs of girth at least 6 are 2-choosable so that each monochromatic component is a path with at most 3 vertices. Our second result states that every planar graph of girth 5 is 2-choosable so that each monochromatic component is a tree with at most 515 vertices. Finally, we prove that every graph with fractional arboricity at most $\frac{2d+2}{d+2}$ is 2-choosabale with the property that each monochromatic component is a tree with maximum degree at most $d$. This implies that planar graphs of girth 5, 6, and 8 are 2-choosable so that each monochromatic component is a tree with maximum degree at most 4, 2, and 1, respectively. All our results are obtained by applying the Nine Dragon Tree Theorem, which was recently proved by Jiang and Yang, and the Strong Nine Dragon Tree Conjecture partially confirmed by Kim et al. and Moore.