{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,19]],"date-time":"2025-09-19T07:55:39Z","timestamp":1758268539372,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>For a graph $G$ and $p\\in [0,1]$, let $G_p$ arise from $G$ by deleting every edge mutually independently with probability $1-p$.\u00a0The random graph model $(K_n)_p$ is certainly the most investigated random graph model and also known as the $G(n,p)$-model.\u00a0We show that several results concerning the length of the longest path\/cycle naturally translate\u00a0to $G_p$ if $G$ is an arbitrary graph of minimum degree at least $n-1$.For a constant $c&gt;0$ and $p=\\frac{c}{n}$, we show that asymptotically almost surely the length of the longest path in $G_p$ is at least $(1-(1+\\epsilon(c))ce^{-c})n$\u00a0for some function $\\epsilon(c)\\to 0$ as $c\\to \\infty$,\u00a0and the length of the longest cycle is a least $(1-O(c^{- \\frac{1}{5}}))n$.\u00a0The first result is asymptotically best-possible.\u00a0This extends several known results on the length of the longest path\/cycle of a random graph in the $G(n,p)$-model to the random graph model $G_p$\u00a0where $G$ is a graph of minimum degree at least $n-1$.<\/jats:p>","DOI":"10.37236\/6761","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T10:33:20Z","timestamp":1578652400000},"source":"Crossref","is-referenced-by-count":3,"title":["Paths and Cycles in Random Subgraphs of Graphs with Large Minimum Degree"],"prefix":"10.37236","volume":"25","author":[{"given":"Stefan","family":"Ehard","sequence":"first","affiliation":[]},{"given":"Felix","family":"Joos","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2018,5,25]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v25i2p31\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v25i2p31\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,16]],"date-time":"2020-01-16T23:33:40Z","timestamp":1579217620000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v25i2p31"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,5,25]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2018,4,13]]}},"URL":"https:\/\/doi.org\/10.37236\/6761","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2018,5,25]]},"article-number":"P2.31"}}