Abstract
Recently, settling a question of Erdős, Balogh, and Petříčková 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–Szemerédi removal lemma, the Erdős–Simonovits stability theorem, and recent results of Balogh, Morris, and Samotij and Saxton and 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.
Highlights
Given a family of combinatorial objects with certain properties, a fundamental problem in extremal combinatorics is to describe the typical structure of these objects
This was initiated in a seminal work of Erdos, Kleitman and Rothschild [13] in 1976
We show an asymptotic version of Theorem 1.1, which implies that almost all maximal triangle-free graphs have a structure very close to the desired one
Summary
Given a family of combinatorial objects with certain properties, a fundamental problem in extremal combinatorics is to describe the typical structure of these objects. Balogh and Petrıckova [11] recently proved a matching upper bound, that the number of maximal triangle-free graphs on vertex set [n] is at most 2n2/8+o(n2). For almost every maximal triangle-free graph G on [n], there is a vertex partition X ∪ Y such that G[X ] is a perfect matching and Y is an independent set.
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