Regular Graphs with Many Triangles are Structured
Name
10369-PDF file-38942-1-10-20220118.pdf
Description
Published version
Size
338.88 KB
Format
Adobe PDF
Checksum (MD5)
fde6cbe2a741a39afaf904b426e2f739
Author(s)
Van der Hoorn, Pim
Lippner, Gabor
Mossel, Elchanan
Date Issued
2022
Journal
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
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.
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
Creative Commons Attribution-NonCommercial-NoDerivs License
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.37236/10369
