The chromatic profile of locally bipartite graphs

In 1973, Erd\H{o}s 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.