The typical structure of maximal triangle-free graphs

TL;DR

This paper characterizes the typical structure of maximal triangle-free graphs, showing that almost all such graphs have a specific vertex partition with a perfect matching and an independent set, using advanced combinatorial tools.

Contribution

It provides a detailed structural description of almost all maximal triangle-free graphs, extending previous bounds and employing new combinatorial bounds and modern hypergraph techniques.

Findings

Almost every maximal triangle-free graph has a vertex partition with a perfect matching and an independent set.

The paper establishes a new bound on the number of maximal independent sets in certain triangle-free graphs.

Utilizes advanced combinatorial theorems to characterize the typical structure of these graphs.

Abstract

Recently, settling a question of Erd\H{o}s, Balogh and Pet\v{r}\'{i}\v{c}kov\'{a} showed that there are at most $2^{n^2/8+o(n^2)}$ $n$-vertex maximal triangle-free graphs, matching the previously known lower bound. Here we characterize the typical structure of maximal triangle-free graphs. We show that almost every maximal triangle-free graph $G$ admits a vertex partition $X\cup Y$ such that $G[X]$ is a perfect matching and $Y$ is an independent set. Our proof uses the Ruzsa-Szemer\'{e}di removal lemma, the Erd\H{o}s-Simonovits stability theorem, and recent results of Balogh-Morris-Samotij and Saxton-Thomason on characterization of the structure of independent sets in hypergraphs. The proof also relies on a new bound on the number of maximal independent sets in triangle-free graphs with many vertex-disjoint $P_3$'s, which is of independent interest.