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 Thomassé. 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 Łuczak and Thomassé, 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.

1973 年，Erdős 和 Simonovits 询问是否每个具有最小度数大于$1/3\cdot n$是 3 色的。这个问题引发了对无三角形图的色度轮廓的研究：对于每个k，什么最小程度可以保证无三角形图是k可着色的。这个问题有着悠久的历史，最终由 Brandt 和 Thomassé 完全解决。对于一般H的无H图的色度分布知之甚少。

无三角形图正是那些每个邻域都是一种可着色的图。由 Łuczak 和 Thomassé 首次提到的局部二部图是无三角形图的自然变体，其中每个邻域都是二部的。在这里，我们研究了局部二分图的色度分布。我们证明了每个n顶点的局部二分图最小度数大于$4/7\cdot n$是 3-colorable (4/7 是紧的) 并且最小程度大于$6/11\cdot n$是 4 色的。尽管局部二分图和无三角形图的色度分布有一些相似之处，但我们会看到存在显着差异。