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.

Erdős, Kleitman, and Rothschild showed that almost every triangle-free graph is bipartite, which leads to an asymptotic formula for the number of triangle-free graphs on n vertices. Osthus, Prömel, 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, which likewise leads to an asymptotic formula for their number. 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 $m=\Theta \left(n^{3/2}\sqrt{\log n}\right)$. 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:

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  Isolated edges, then trees, then more complex subgraphs emerge as ‘defect edges’, the edges within the parts of a max cut of G. In fact, the distribution of defect edges is first that of independent Erdős-Rényi random graphs inside the parts, then that of independent exponential random graphs, conditioned on a small maximum degree and no triangles.
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  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 $m\sim \frac{1}{4}{n}^{3/2}\sqrt{\log n}$.
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  Giant components emerge in the defect edges at $m\sim \frac{1}{4}{n}^{3/2}\sqrt{\log n}$.

We further use this structural characterization to prove asymptotic formulas for the number of triangle-free graphs with n vertices and m edges in this range of densities. The asymptotic formula exhibits a change in form around the threshold $m\sim \frac{1}{4}{n}^{3/2}\sqrt{\log n}$ at which giant components emerge among the defect edges.

We likewise prove the analogous results for the random graph $G(n,p)$ conditioned on triangle-freeness.