THE TYPICAL STRUCTURE OF MAXIMAL TRIANGLE-FREE GRAPHS - JÓZSEF BALOGH; HONG LIU; ŠÁRKA PETŘÍČKOVÁ; MARYAM SHARIFZADEH (Forum of Mathematics, Sigma, 2015)
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.