Minimum degree stability of \(H\)-free graphs

• Published in: arXiv.org
• Main Author: Illingworth, Freddie
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 21.08.2023
• Subjects: Graph theory; Graphs; Stability
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.2102.11104

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 Andr\'{a}sfai-Erdős-S\'{o}s theorem and work of Alon and Sudakov.