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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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
Date Deposited:	05 Apr 2018 10:07
Last Modified:	09 Nov 2024 22:39
URI:	http://eprints.lse.ac.uk/id/eprint/87369

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