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Title: 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.  more » « less
Award ID(s):
1800738 1741355
PAR ID:
10350674
Author(s) / Creator(s):
; ;
Date Published:
Journal Name:
The Electronic Journal of Combinatorics
Volume:
29
Issue:
1
ISSN:
1077-8926
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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