Every n-vertex H-minor-free graph admits a 3-coloring with monochromatic components of size O_H(n^{4/9}).
Kleinberg, Rajeev Motwani, Prabhakar Raghavan, and Suresh Venkatasubrama- nian
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3-Colouring Graphs Excluding a Fixed Minor
Every n-vertex H-minor-free graph admits a 3-coloring with monochromatic components of size O_H(n^{4/9}).