Regular Graphs with Many Triangles are Structured
Author(s)
Van der Hoorn, Pim; Lippner, Gabor; Mossel, Elchanan
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<jats:p>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.</jats:p>
Date issued
2022Department
Massachusetts Institute of Technology. Department of MathematicsJournal
The Electronic Journal of Combinatorics
Publisher
The Electronic Journal of Combinatorics
Citation
Van der Hoorn, Pim, Lippner, Gabor and Mossel, Elchanan. 2022. "Regular Graphs with Many Triangles are Structured." The Electronic Journal of Combinatorics, 29 (1).
Version: Final published version