Abstract
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.
Highlights
What is the probability that a random graph has a lot more triangles than expected? This is a typical question in the field of large deviations, the theory that studies the tail behavior of random variables or, stated differently, the behavior of random objects conditioned on a parameter being far from its expectation
Our results further point in this direction: we show that typical d-regular graphs with many triangles contain a large, highly structured, part
Motivated by the question “can local constraints induce global behavior?”, we study the random d-regular graph Gd(n) conditioned on having at least a positive fraction of the maximum possible number of triangles. (For d fixed this just means linearly many triangles, in n.) With respect to the previous section, our setting is related to the entropy maximization problem with local and global constraints, i.e. where each node must have degree exactly d and must be incident to at least t triangles on average
Summary
In this paper we prove a strong, almost sure structural stability result for this extremal problem: Let 0 < c 1 and let Gd,c(n) denote the set of d-regular graphs on n labeled nodes that contain at least c · Tmax triangles. For constant d and large n, almost all graphs in Gd,c(n) consist of a disjoint union of (d + 1)-cliques and an almost triangle-free part This result may not seem surprising at first, and especially for c close to 1 it seems quite natural to expect. We prove that the observation that triangles are clustered in typical elements of Gd,c(n) holds in a very strong sense: for almost all graphs in the class, almost all triangles are contained in (d + 1)-cliques
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