Minimum Degree Stability of H-Free Graphs

Abstract

Given an (r + 1)-chromatic graph H, the fundamental edge stability result of Erdős and Simonovits says that all n-vertex H-free graphs have at most (1 - 1/r + o(1)) binom{n}{2} edges, and any H-free graph with that many edges can be made r-partite by deleting o(n^{2}) edges. Here we consider a natural variant of this—the minimum degree stability of H-free graphs. In particular, what is the least c such that any n-vertex H-free graph with minimum degree greater than cn can be made r-partite by deleting o(n^{2}) edges? We determine this least value for all 3-chromatic H and for very many non-3-colourable H (all those in which one is commonly interested) as well as bounding it for the remainder. This extends the Andrásfai-Erdős-Sós theorem and work of Alon and Sudakov.