The chromatic profile of locally bipartite graphs

• Published in: arXiv.org
• Main Author: Illingworth, Freddie
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 21.08.2023
• Subjects: Binary system; Graph theory; Graphs; Mathematics - Combinatorics
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.2012.10409

In 1973, Erdős and Simonovits asked whether every \(n\)-vertex triangle-free graph with minimum degree greater than \(1/3 \cdot n\) is 3-colourable. This question initiated the study of the chromatic profile of triangle-free graphs: for each \(k\), what minimum degree guarantees that a triangle-free graph is \(k\)-colourable. This problem has a rich history which culminated in its complete solution by Brandt and Thomass\'{e}. Much less is known about the chromatic profile of \(H\)-free graphs for general \(H\). Triangle-free graphs are exactly those in which each neighbourhood is one-colourable. Locally bipartite graphs, first mentioned by Luczak and Thomass\'{e}, are the natural variant of triangle-free graphs in which each neighbourhood is bipartite. Here we study the chromatic profile of locally bipartite graphs. We show that every \(n\)-vertex locally bipartite graph with minimum degree greater than \(4/7 \cdot n\) is 3-colourable (\(4/7\) is tight) and with minimum degree greater than \(6/11 \cdot n\) is 4-colourable. Although the chromatic profiles of locally bipartite and triangle-free graphs bear some similarities, we will see there are striking differences.