The chromatic profile of locally colourable graphs

• Published in: Combinatorics, probability & computing Vol. 31; no. 6; pp. 976 - 1009
• Main Author: Illingworth, Freddie
• Format: Journal Article
• Language: English
• Published: Cambridge Cambridge University Press 01.11.2022
• Subjects: Graph theory; Graphs; Neighborhoods; Theorems
• ISSN: 0963-5483; 1469-2163
• DOI: 10.1017/S0963548322000050

Abstract The classical Andrásfai-Erdős-Sós theorem considers the chromatic number of $K_{r + 1}$ -free graphs with large minimum degree, and in the case, $r = 2$ says that any n -vertex triangle-free graph with minimum degree greater than $2/5 \cdot n$ is bipartite. This began 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? The chromatic profile has been extensively studied and was finally determined by Brandt and Thomassé. Triangle-free graphs are exactly those in which each neighbourhood is one-colourable. As a natural variant, Luczak and Thomassé introduced the notion of a locally bipartite graph in which each neighbourhood is 2-colourable. Here we study the chromatic profile of the family of graphs in which every neighbourhood is b -colourable (locally b -partite graphs) as well as the family where the common neighbourhood of every a -clique is b -colourable. Our results include the chromatic thresholds of these families (extending a result of Allen, Böttcher, Griffiths, Kohayakawa and Morris) as well as showing that every n -vertex locally b -partite graph with minimum degree greater than $(1 - 1/(b + 1/7)) \cdot n$ is $(b + 1)$ -colourable. Understanding these locally colourable graphs is crucial for extending the Andrásfai-Erdős-Sós theorem to non-complete graphs, which we develop elsewhere.