Minimum degree stability of $H$-free graphs

TL;DR

This paper investigates the minimum degree conditions under which $H$-free graphs can be made $r$-partite by removing few edges, extending classical stability results to a new setting.

Contribution

It determines the least minimum degree threshold for $H$-free graphs to be close to $r$-partite, covering all 3-chromatic graphs and many non-3-colorable graphs.

Findings

Established the minimum degree threshold for all 3-chromatic $H$.

Extended stability results to minimum degree conditions.

Provided bounds for the remaining non-3-colorable graphs.

Abstract

Given an $(r + 1)$-chromatic graph $H$, the fundamental edge stability result of Erd\H{o}s 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-Erd\H{o}s-S\'{o}s theorem and work of Alon and Sudakov.