Clustered Colouring in Minor-Closed Classes

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 (2t+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.

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Acknowledgments

This research was initiated at the 2017 Barbados Graph Theory Workshop held at the Bellairs Research Institute. Thanks to the workshop participants for creating a stimulating working environment. Thanks to the referees for several instructive comments.

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Correspondence to Sergey Norin or Alex Scott or Paul Seymour or David R. Wood.

Additional information

Supported by NSERC grant 418520.

Supported by a Leverhulme Trust Research Fellowship.

Supported by ONR grant N00014-14-1-0084 and NSF grant DMS-1265563.

Supported by the Australian Research Council.

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Norin, S., Scott, A., Seymour, P. et al. Clustered Colouring in Minor-Closed Classes. Combinatorica 39, 1387–1412 (2019). https://doi.org/10.1007/s00493-019-3848-z

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Mathematics Subject Classification (2010)

  • 05C83
  • 05C15