When are random regular triangle-free graphs bipartite?

Abstract

Let $\mathcal T_{n,m}$ denote the set of triangle-free graphs on $n$ vertices with $m$ edges where $m = m(n)$. A natural and well-studied question in random graph theory is ``for large $n$, how large is $\mathcal T_{n,m}$?" This question has been answered when $m \le n^{3/2-\eps}$ for some $\eps>0$, or $m \geq \frac{13}{56}n^{3/2}\sqrt{\log n}$. These results depend on the typical structure of graphs in $\mathcal T_{n,m}$, where in the sparse regime these graphs are unstructured and in the dense regime these graphs are rigidly structured. One such result is that $m = \frac{\sqrt 3}{4}n^{3/2}\sqrt{\log n}$ is the tight threshold for a graph chosen uniformly at random from $\mathcal T_{n,m}$ to be bipartite.

Another popular topic in random graph theory is the study of random regular graphs. Let $G'$ be random graph chosen uniformly from all $d$-regular graphs on $n$ vertices. Let $X$ be the number of triangles in $G'$. One particular question that has been considered recently is ``what is the distribution of $X$?'' The answer to this question is well understood when $X$ is near its mean for certain values of $d$, but is not well understood in the lower or upper tail of the distribution; in particular, when $X=0$.

In this thesis, we establish that $d = \frac{\sqrt 3}{2} \sqrt{n\log n}$ is the sharp threshold for random $d$-regular triangle-free graphs being bipartite, which implies an asymptotic enumeration of $d$-regular triangle-free graphs when $(1 + \eps)\frac{\sqrt 3}{2}\sqrt{n \log n} \leq d \leq \mu_0 n$ for any $\eps > 0$ and sufficiently small $\mu_0 > 0$ (the restriction of $d\le \mu_0 n$ is due to some technical considerations). This extends the enumeration of triangle-free graphs to the regular setting and broadens our understanding of the extreme lower tail of the distribution of the number of triangles in a random regular graph.