Abstract

We tie together two natural but, a priori, different themes. As a starting point consider Erdős and Simonovits’s classical edge stability for an (r + 1)-chromatic graph H. This says that any n-vertex H-free graph with (1 − 1/r + o(1))*(n choose 2) edges is close to (within o(n^2) edges of) r-partite. This is false if 1 − 1/r is replaced by any smaller constant. However, instead of insisting on many edges, what if we ask that the n-vertex graph has large minimum degree? This is the basic question of minimum degree stability: what constant c guarantees that any n-vertex H-free graph with minimum degree greater than cn is close to r-partite? c depends not just on chromatic number of H but also on its finer structure.

Somewhat surprisingly, answering the minimum degree stability question requires understanding locally colourable graphs—graphs in which every neighbourhood has small chromatic number—with large minimum degree. This is a natural local-to-global colouring question: if every neighbourhood is big and has small chromatic number must the whole graph have small chromatic number? The triangle-free case has a rich history. The more general case has some similarities but also striking differences.