Graph coloring with no large monochromatic components

For a graph G and an integer t we let m c c_t (G) be the smallest m such that there exists a coloring of the vertices of G by t colors with no monochromatic connected subgraph having more than m vertices. Let F be any nontrivial minor-closed family of graphs. We show that mcc₂ (G) = O (n^{2 / 3}) for any n-vertex graph G ∈ F. This bound is asymptotically optimal and it is attained for planar graphs. More generally, for every such F, and every fixed t we show that mcc_t (G) = O (n^{2 / (t + 1)}). On the other hand, we have examples of graphs G with no K_{t + 3} minor and with mcc_t (G) = Ω (n^{(2 / 2 t - 1)}). It is also interesting to consider graphs of bounded degrees. Haxell, Szabó, and Tardos proved mcc₂ (G) ≤ 20000 for every graph G of maximum degree 5. We show that there are n-vertex 7-regular graphs G with mcc₂ (G) = Ω (n), and more sharply, for every ε > 0 there exists c_ε > 0 and n-vertex graphs of maximum degree 7, average degree at most 6 + ε for all subgraphs, and with mcc₂ (G) ≥ c_ε n. For 6-regular graphs it is known only that the maximum order of magnitude of mcc₂ is between sqrt(n) and n. We also offer a Ramsey-theoretic perspective of the quantity m c c_t (G).