Chromatic number is not tournament-local

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B Pub Date : 2024-05-10 DOI:10.1016/j.jctb.2024.04.005

António Girão , Kevin Hendrey , Freddie Illingworth , Florian Lehner , Lukas Michel , Michael Savery , Raphael Steiner

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引用次数: 0

Abstract

Scott and Seymour conjectured the existence of a function $f : N \to N$ such that, for every graph G and tournament T on the same vertex set, $χ (G) ⩾ f (k)$ implies that $χ (G [N_{T}^{+} (v)]) ⩾ k$ 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.

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色度编号不是赛事本地编号

斯科特（Scott）和西摩（Seymour）猜想存在这样一个函数 f:N→N：对于每个图 G 和同一顶点集上的锦标赛 T，χ(G)⩾f(k)意味着对于某个顶点 v，χ(G[NT+(v)])⩾k。在本注释中，我们推翻了这一猜想，即使 v 被大小为 O(log|V(G)|)的顶点集所替代。因此，我们否定地回答了 Harutyunyan、Le、Thomassé 和 Wu 提出的关于图 G 被另一个锦标赛取代时的相应问题，并推翻了 Nguyen、Scott 和 Seymour 提出的相关猜想。我们还证明，在色度数被退化性取代的情况下，会表现出截然不同的行为。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Combinatorial Theory Series B 数学-数学

CiteScore

2.70

自引率

14.30%

发文量

审稿时长

6-12 weeks

期刊介绍： The Journal of Combinatorial Theory publishes original mathematical research dealing with theoretical and physical aspects of the study of finite and discrete structures in all branches of science. Series B is concerned primarily with graph theory and matroid theory and is a valuable tool for mathematicians and computer scientists.