Illingworth, Freddie; (2022) The chromatic profile of locally colourable graphs. Combinatorics, Probability and Computing , 31 (6) pp. 976-1009. 10.1017/S0963548322000050.

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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 anyn-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 eachk, what minimum degree guarantees that a triangle-free graph isk-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 isb-colourable (locallyb-partite graphs) as well as the family where the common neighbourhood of everya-clique isb-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 everyn-vertex locallyb-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.

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