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Award ID contains: 1800738

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  1. ; ; (, The Electronic Journal of Combinatorics)
    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. 
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  2. ; ; ; ; (, Physical Review Research)
    null (Ed.)
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  3. ; ; ; ; (, Algebraic Combinatorics)
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