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.