Regular Graphs with Many Triangles are Structured
We compute the leading asymptotics of the logarithm of the number of $d$-regular graphs having at least a fixed positive fraction $c$ of the maximum possible number of triangles, and provide a strong structural description of almost all such graphs.
When $d$ is constant, 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 very dense sets of size $d+o(d)$ 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.