### Abstract

The clustered chromatic number of a class of graphs is the minimum integer k such that for some integer c every graph in the class is k-colourable with monochromatic components of size at most c. We prove that for every graph H, the clustered chromatic number of the class of H-minor-free graphs is tied to the tree-depth of H. In particular, if H is connected with tree-depth t, then every H-minor-free graph is (2^{t+1}–4)-colourable with monochromatic components of size at most c(H). This provides the first evidence for a conjecture of Ossona de Mendez, Oum and Wood (2016) about defective colouring of H-minor-free graphs. If t = 3, then we prove that 4 colours suffie, which is best possible. We also determine those minor-closed graph classes with clustered chromatic number 2. Finally, we develop a conjecture for the clustered chromatic number of an arbitrary minor-closed class.

Original language | English (US) |
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Pages (from-to) | 1387-1412 |

Number of pages | 26 |

Journal | Combinatorica |

Volume | 39 |

Issue number | 6 |

DOIs | |

State | Published - Dec 1 2019 |

### All Science Journal Classification (ASJC) codes

- Discrete Mathematics and Combinatorics
- Computational Mathematics

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## Cite this

*Combinatorica*,

*39*(6), 1387-1412. https://doi.org/10.1007/s00493-019-3848-z