Regular graphs with linearly many triangles

Abstract

A $d$-regular graph on $n$ nodes has at most $T_{\max }=\frac{n}{3}\left(\genfrac{}{}{0ex}{}{d}{2}\right)$ triangles. We compute the leading asymptotics of the probability that a large random $d$-regular graph has at least $c\cdot T_{\max }$ triangles, and provide a strong structural description of such graphs. When $d$ is fixed, we show that such graphs typically consist of many disjoint $d+1$-cliques and an almost triangle-free part. When $d$ is allowed to grow with $n$, we show that such graphs typically consist of $d+o(d)$ sized almost cliques together with an almost triangle-free part. This confirms a conjecture of Collet and Eckmann from 2002 and considerably strengthens their observation that the triangles cannot be totally scattered in typical instances of regular graphs with many triangles.

Added extra context of the results via a new reference (Collet, Eckmann, 2002)