A Note on Stability for Maximal F-Free Graphs

Dániel Gerbner

doi:10.1007/s00373-021-02372-z

Abstract

Popielarz, Sahasrabuddhe and Snyder in 2018 proved that maximal K_{r+1}-free graphs with (1-frac{1}{r})frac{n^2}{2}-o(n^{frac{r+1}{r}}) edges contain a complete r-partite subgraph on n-o(n) vertices. This was very recently extended to odd cycles in place of K_3 by Wang, Wang, Yang and Yuan. We further extend it to some other 3-chromatic graphs, and obtain some other stability results along the way.

Highlights

This was very recently extended to odd cycles in place of K3 by Wang, Wang, Yang and Yuan. We further extend it to some other 3-chromatic graphs, and obtain some other stability results along the way
One of the most basic questions of graph theory is the following: given a graph F, how many edges can an n-vertex graph G have if it is F-free, i.e. G does not contain F as a subgraph? This quantity is denoted by exðn; FÞ
This phenomenon is called stability and there are several non-equivalent stability theorems concerning the same graphs, where the differences come from the precise form of ‘‘almost’’, ‘‘structure’’ and ‘‘very similar’’ in the previous sentence

Summary

Introduction

One of the most basic questions of graph theory is the following: given a graph F, how many edges can an n-vertex graph G have if it is F-free, i.e. G does not contain F as a subgraph? This quantity is denoted by exðn; FÞ. Consider a Krþ1-saturated graph with close to trðnÞ edges Does it contain a large complete r-partite subgraph? Every Krþ1-saturated graph G on n vertices with trðnÞ À oðnrþr1Þ edges contains a complete r-partite subgraph on ð1 À oð1ÞÞn vertices. There are Krþ saturated graphs on n vertices with trðnÞ À Xðnrþr1Þ edges that do not contain a complete r-partite subgraph on ð1 À oð1ÞÞn vertices. We will prove Conjecture 1.4 for some 3-chromatic graphs with a color-critical edge, but we will use another lemma instead, making this paper self-contained. Theorem 1.6 Let F be a 3-chromatic graph with a color-critical edge in which every edge has a vertex that is contained in a triangle.

Lemmas and Other Results

Proofs