Every n-vertex planar graph can be 3-coloured so that each monochromatic connected component has at most O(n^{4/9}) vertices, improving the previous O(n^{1/2}) bound.
Weak diameter choosability of graphs with an excluded minor
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3-Colouring Planar Graphs
Every n-vertex planar graph can be 3-coloured so that each monochromatic connected component has at most O(n^{4/9}) vertices, improving the previous O(n^{1/2}) bound.