{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:37Z","timestamp":1753893817802,"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>Let $\\mathcal{H}$ be a given finite (possibly empty) family of connected graphs, each containing a cycle, and let $G$ be an arbitrary finite $\\mathcal{H}$-free graph with minimum degree at least $k$. For $p \\in [0,1]$, we form a $p$-random subgraph $G_p$ of $G$ by independently keeping each edge of $G$ with probability $p$. Extending a classical result of Ajtai, Koml\u00f3s, and Szemer\u00e9di, we prove that for every positive $\\varepsilon$, there exists a positive $\\delta$ (depending only on $\\varepsilon$) such that the following holds: If $p \\geq \\frac{1+\\varepsilon}{k}$, then with probability tending to $1$ as $k \\to \\infty$, the random graph $G_p$ contains a cycle of length at least $n_{\\mathcal{H}}(\\delta k)$, where $n_\\mathcal{H}(k)&gt;k$ is the minimum number of vertices in an $\\mathcal{H}$-free graph of average degree at least $k$. Thus in particular $G_p$ as above typically contains a cycle of length at least linear in $k$.<\/jats:p>","DOI":"10.37236\/3198","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T01:10:00Z","timestamp":1578705000000},"source":"Crossref","is-referenced-by-count":4,"title":["Long Paths and Cycles in Random Subgraphs of $\\mathcal{H}$-Free Graphs"],"prefix":"10.37236","volume":"21","author":[{"given":"Michael","family":"Krivelevich","sequence":"first","affiliation":[]},{"given":"Wojciech","family":"Samotij","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2014,2,13]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v21i1p30\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v21i1p30\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T11:05:23Z","timestamp":1579259123000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v21i1p30"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,2,13]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2014,1,13]]}},"URL":"https:\/\/doi.org\/10.37236\/3198","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2014,2,13]]},"article-number":"P1.30"}}