{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,26]],"date-time":"2025-11-26T16:07:26Z","timestamp":1764173246520,"version":"3.41.0"},"reference-count":0,"publisher":"Cambridge University Press (CUP)","issue":"1-2","license":[{"start":{"date-parts":[[2005,2,15]],"date-time":"2005-02-15T00:00:00Z","timestamp":1108425600000},"content-version":"unspecified","delay-in-days":45,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2005,1]]},"abstract":"<jats:p>Let <jats:inline-formula>$G$<\/jats:inline-formula> be a noncomplete <jats:inline-formula>$k$<\/jats:inline-formula>-connected graph such that the graphs obtained from contracting any edge in <jats:inline-formula>$G$<\/jats:inline-formula> are not <jats:inline-formula>$k$<\/jats:inline-formula>-connected, and let <jats:inline-formula>$t(G)$<\/jats:inline-formula> denote the number of triangles in <jats:inline-formula>$G$<\/jats:inline-formula>. Thomassen proved <jats:inline-formula>$t(G) \\geq 1$<\/jats:inline-formula>, which was later improved by Mader to <jats:inline-formula>$t(G) \\geq \\frac{1}{3}|V(G)|$<\/jats:inline-formula>.<\/jats:p>\n\t  <jats:p>Here we show <jats:inline-formula>$t(G) \\geq \\frac{2}{3}|V(G)|$<\/jats:inline-formula> (which is best possible in general).<\/jats:p>\n\t  <jats:p>Furthermore it is proved that, for <jats:inline-formula>$k \\geq 4$<\/jats:inline-formula>, a <jats:inline-formula>$k$<\/jats:inline-formula>-connected graph without two disjoint triangles must contain an edge not contained in a triangle whose contraction yields a <jats:inline-formula>$k$<\/jats:inline-formula>-connected graph. As an application, for <jats:inline-formula>$k \\geq 4$<\/jats:inline-formula> every <jats:inline-formula>$k$<\/jats:inline-formula>-connected graph <jats:inline-formula>$G$<\/jats:inline-formula> admits two disjoint induced cycles <jats:inline-formula>$C_1,C_2$<\/jats:inline-formula> such that <jats:inline-formula>$G-V(C_1)$<\/jats:inline-formula> and <jats:inline-formula>$G-V(C_2)$<\/jats:inline-formula> are <jats:inline-formula>$(k-3)$<\/jats:inline-formula>-connected.<\/jats:p>","DOI":"10.1017\/s0963548304006601","type":"journal-article","created":{"date-parts":[[2005,2,15]],"date-time":"2005-02-15T12:56:08Z","timestamp":1108472168000},"page":"133-146","source":"Crossref","is-referenced-by-count":7,"title":["Triangle Density and Contractibility"],"prefix":"10.1017","volume":"14","author":[{"given":"MATTHIAS","family":"KRIESELL","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2005,2,15]]},"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548304006601","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,20]],"date-time":"2025-06-20T21:15:42Z","timestamp":1750454142000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548304006601\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2005,1]]},"references-count":0,"journal-issue":{"issue":"1-2","published-print":{"date-parts":[[2005,7]]}},"alternative-id":["S0963548304006601"],"URL":"https:\/\/doi.org\/10.1017\/s0963548304006601","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"type":"print","value":"0963-5483"},{"type":"electronic","value":"1469-2163"}],"subject":[],"published":{"date-parts":[[2005,1]]}}}