## Abstract

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 mcc_{2} (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 mcc_{2} (G) ≤ 20000 for every graph G of maximum degree 5. We show that there are n-vertex 7-regular graphs G with mcc_{2} (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 mcc_{2} (G) ≥ c_{ε} n. For 6-regular graphs it is known only that the maximum order of magnitude of mcc_{2} is between sqrt(n) and n. We also offer a Ramsey-theoretic perspective of the quantity m c c_{t} (G).

Original language | American English |
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Pages (from-to) | 115-122 |

Number of pages | 8 |

Journal | Electronic Notes in Discrete Mathematics |

Volume | 29 |

Issue number | SPEC. ISS. |

DOIs | |

State | Published - 15 Aug 2007 |