A note on stability for maximal $F$-free graphs

TL;DR

This paper extends stability results for maximal $F$-free graphs, demonstrating that such graphs with near-maximum edges contain large structured subgraphs, generalizing previous results for complete graphs and odd cycles.

Contribution

It broadens stability theorems to include additional 3-chromatic graphs beyond cycles and complete graphs, enhancing understanding of graph structure near extremal edge counts.

Findings

Maximal $F$-free graphs with many edges contain large structured subgraphs.

Extended stability results to new classes of 3-chromatic graphs.

Provided new stability theorems along the way.

Abstract

Popielarz, Sahasrabudhe 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.