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arXiv · 1904.02212

Regular graphs with linearly many triangles

Pim van der Hoorn·Gabor Lippner·Elchanan Mossel
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Abstract

A $d$-regular graph on $n$ nodes has at most $T_{\max} = \frac{n}{3} \tbinom{d}{2}$ 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.

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Pim van der Hoorn, Gabor Lippner, Elchanan Mossel. 2019-04-03. Regular graphs with linearly many triangles. https://arxiv.org/abs/1904.02212

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