van den Heuvel, Jan 
ORCID: 0000-0003-0897-9148, Ossona de Mendez, Patrice, Quiroz, Daniel, Rabinovich, Roman and Siebertz, Sebastian
(2017)
On the generalised colouring numbers of graphs that exclude a fixed minor.
European Journal of Combinatorics, 66.
pp. 129-144.
ISSN 0195-6698
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Abstract
The generalised colouring numbers colr(G) and wcolr(G) were introduced by Kierstead and Yang as a generalisation of the usual colouring number, and have since then found important theoretical and algorithmic applications. In this paper, we dramatically improve upon the known upper bounds for generalised colouring numbers for graphs excluding a fixed minor, from the exponential bounds of Grohe et al. to a linear bound for the r-colouring number colr and a polynomial bound for the weak r-colouring number wcolr. In particular, we show that if G excludes Kt as a minor, for some fixed t≥4, then colr(G)≤(t−12)(2r+1) and wcolr(G)≤(r+t−2t−2)⋅(t−3)(2r+1)∈O(rt−1). In the case of graphs G of bounded genus g, we improve the bounds to colr(G)≤(2g+3)(2r+1) (and even colr(G)≤5r+1 if g=0, i.e. if G is planar) and wcolr(G)≤(2g+(r+22))(2r+1).
| Item Type: | Article |
|---|---|
| Official URL: | https://www.journals.elsevier.com/european-journal... |
| Additional Information: | © 2017 Elsevier Ltd. |
| Divisions: | Mathematics |
| Subjects: | Q Science > QA Mathematics |
| Date Deposited: | 17 Jul 2017 14:10 |
| Last Modified: | 12 Dec 2024 01:31 |
| URI: | http://eprints.lse.ac.uk/id/eprint/83557 |
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