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Improper colourings inspired by Hadwiger’s conjecture

van den Heuvel, Jan ORCID: 0000-0003-0897-9148 and Wood, David R. (2018) Improper colourings inspired by Hadwiger’s conjecture. Journal of the London Mathematical Society, 98 (1). 129 - 148. ISSN 0024-6107

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Identification Number: 10.1112/jlms.12127

Abstract

Hadwiger’s Conjecture asserts that every Kt-minor-free graph has a proper (t − 1)-colouring. We relax the conclusion in Hadwiger’s Conjecture via improper colourings. We prove that every Kt-minor-free graph is (2t − 2)-colourable with monochromatic components of order at most 1/2 (t − 2). This result has no more colours and much smaller monochromatic components than all previous results in this direction. We then prove that every Kt-minor-free graph is (t − 1)-colourable with monochromatic degree at most t − 2. This is the best known degree bound for such a result. Both these theorems are based on a decomposition method of independent interest. We give analogous results for Ks,t-minorfree graphs, which lead to improved bounds on generalised colouring numbers for these classes. Finally, we prove that graphs containing no Kt-immersion are 2-colourable with bounded monochromatic degree.

Item Type: Article
Official URL: https://www.lms.ac.uk/publications/jlms
Additional Information: © 2018 Oxford University Press on behalf of the London Mathematical Society
Divisions: Mathematics
Subjects: Q Science > QA Mathematics
Sets: Departments > Mathematics
Date Deposited: 05 Apr 2018 10:07
Last Modified: 20 Oct 2021 01:25
URI: http://eprints.lse.ac.uk/id/eprint/87369

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