### 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 | English |
---|---|

Pages (from-to) | 115-122 |

Number of pages | 8 |

Journal | Electronic Notes in Discrete Mathematics |

Volume | 29 |

Issue number | SPEC. ISS. |

DOIs | |

Publication status | Published - Aug 15 2007 |

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### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics

### Cite this

*Electronic Notes in Discrete Mathematics*,

*29*(SPEC. ISS.), 115-122. https://doi.org/10.1016/j.endm.2007.07.020

**Graph coloring with no large monochromatic components.** / Linial, Nathan; Matoušek, Jiří; Sheffet, Or; Tardos, G.

Research output: Contribution to journal › Article

*Electronic Notes in Discrete Mathematics*, vol. 29, no. SPEC. ISS., pp. 115-122. https://doi.org/10.1016/j.endm.2007.07.020

}

TY - JOUR

T1 - Graph coloring with no large monochromatic components

AU - Linial, Nathan

AU - Matoušek, Jiří

AU - Sheffet, Or

AU - Tardos, G.

PY - 2007/8/15

Y1 - 2007/8/15

N2 - For a graph G and an integer t we let m c ct (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 (n2 / 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 mcct (G) = O (n2 / (t + 1)). On the other hand, we have examples of graphs G with no Kt + 3 minor and with mcct (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 cε > 0 and n-vertex graphs of maximum degree 7, average degree at most 6 + ε for all subgraphs, and with mcc2 (G) ≥ cε 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 ct (G).

AB - For a graph G and an integer t we let m c ct (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 (n2 / 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 mcct (G) = O (n2 / (t + 1)). On the other hand, we have examples of graphs G with no Kt + 3 minor and with mcct (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 cε > 0 and n-vertex graphs of maximum degree 7, average degree at most 6 + ε for all subgraphs, and with mcc2 (G) ≥ cε 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 ct (G).

UR - http://www.scopus.com/inward/record.url?scp=34547761277&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34547761277&partnerID=8YFLogxK

U2 - 10.1016/j.endm.2007.07.020

DO - 10.1016/j.endm.2007.07.020

M3 - Article

VL - 29

SP - 115

EP - 122

JO - Electronic Notes in Discrete Mathematics

JF - Electronic Notes in Discrete Mathematics

SN - 1571-0653

IS - SPEC. ISS.

ER -