Girao, Antonio;
Hendrey, Kevin;
Illingworth, Freddie;
Lehner, Florian;
Michel, Lukas;
Savery, Michael;
Steiner, Raphael;
(2024)
Chromatic number is not tournament-local.
Journal of Combinatorial Theory, Series B
, 168
pp. 86-95.
10.1016/j.jctb.2024.04.005.
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Abstract
Scott and Seymour conjectured the existence of a function f:N→N such that, for every graph G and tournament T on the same vertex set, χ(G)⩾f(k) implies that χ(G[NT+(v)])⩾k for some vertex v. In this note we disprove this conjecture even if v is replaced by a vertex set of size O(log|V(G)|). As a consequence, we answer in the negative a question of Harutyunyan, Le, Thomassé, and Wu concerning the corresponding statement where the graph G is replaced by another tournament, and disprove a related conjecture of Nguyen, Scott, and Seymour. We also show that the setting where chromatic number is replaced by degeneracy exhibits a quite different behaviour.
| Type: | Article |
|---|---|
| Title: | Chromatic number is not tournament-local |
| Open access status: | An open access version is available from UCL Discovery |
| DOI: | 10.1016/j.jctb.2024.04.005 |
| Publisher version: | https://doi.org/10.1016/j.jctb.2024.04.005 |
| Language: | English |
| Additional information: | © 2024 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/). |
| Keywords: | Chromatic number, Tournaments, Schrijver graphs, Degeneracy |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics |
| URI: | https://discovery.ucl.ac.uk/id/eprint/10197007 |
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