Clustered Colouring in Minor-Closed Classes

Combinatorica volume 39, pages1387–1412(2019)Cite this article

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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Affiliations

  1. Department of Mathematics and Statistics, McGill University, Montréal, Canada

    Sergey Norin

  2. Mathematical Institute, University of Oxford, Oxford, UK

    Alex Scott

  3. Department of Mathematics, Princeton University, New Jersey, USA

    Paul Seymour

  4. School of Mathematics, Monash University, Melbourne, Australia

    David R. Wood

Authors
  1. Sergey Norin
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  2. Alex Scott
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  3. Paul Seymour
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  4. David R. Wood
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Corresponding authors

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