{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:49Z","timestamp":1753893829701,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<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.\r\nWhen $d$ 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.\r\nThis 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>","DOI":"10.37236\/10369","type":"journal-article","created":{"date-parts":[[2022,2,2]],"date-time":"2022-02-02T03:11:11Z","timestamp":1643771471000},"source":"Crossref","is-referenced-by-count":1,"title":["Regular Graphs with Many Triangles are Structured"],"prefix":"10.37236","volume":"29","author":[{"given":"Pim","family":"Van der Hoorn","sequence":"first","affiliation":[]},{"given":"Gabor","family":"Lippner","sequence":"additional","affiliation":[]},{"given":"Elchanan","family":"Mossel","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2022,1,28]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v29i1p7\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v29i1p7\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,2,2]],"date-time":"2022-02-02T03:12:23Z","timestamp":1643771543000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v29i1p7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,1,28]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2022,1,27]]}},"URL":"https:\/\/doi.org\/10.37236\/10369","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2022,1,28]]},"article-number":"P1.7"}}