On the evolution of structure in triangle-free graphs

Abstract

We study the typical structure and the number of triangle-free graphs with $n$ vertices and $m$ edges where $m$ is large enough so that a typical triangle-free graph has a cut containing nearly all of its edges, but may not be bipartite. Erd\H{o}s, Kleitman, and Rothschild showed that almost every triangle-free graph is bipartite. Osthus, Pr\"omel, and Taraz later showed that for $m \ge (1+\epsilon)\frac{\sqrt{3}}{4}n^{3/2}\sqrt{\log n}$, almost every triangle-free graph on $n$ vertices and $m$ edges is bipartite. Here we give a precise characterization of the distribution of edges within each part of the max cut of a uniformly chosen triangle-free graph $G$ on $n$ vertices and $m$ edges, for a larger range of densities with $m=\Theta(n^{3/2} \sqrt{\log n})$. Using this characterization, we describe the evolution of the structure of typical triangle-free graphs as the density changes. We show that as the number of edges decreases below $\frac{\sqrt{3}}{4} n^{3/2}\sqrt{\log n}$, the following structural changes occur in $G$: -Isolated edges, then trees, then more complex subgraphs emerge as `defect edges', edges within parts of a max cut of $G$. The distribution of defect edges is first that of independent Erd\H{o}s-R\'{e}nyi random graphs, then that of independent exponential random graphs, conditioned on a small maximum degree and no triangles. -There is a sharp threshold for $3$-colorability at $m \sim \frac{\sqrt{2}}{4} n^{3/2}\sqrt{\log n}$ and a sharp threshold between $4$-colorability and unbounded chromatic number at $m\sim\frac{1}{4}n^{3/2}\sqrt{\log n}$. -Giant components emerge in the defect edges at $m\sim\frac{1}{4} n^{3/2}\sqrt{\log n}$. We use these results to prove asymptotic formulas for the number of triangle-free graphs at these densities. We likewise prove analogous results for the random graph $G(n,p)$ conditioned on triangle-freeness.