{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:56Z","timestamp":1753893836099,"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>Amit and Linial have shown that a random lift of a connected graph with minimum degree $\\delta\\ge3$ is asymptotically almost surely (a.a.s.) $\\delta$-connected and mentioned the problem of estimating this probability as a function of the degree of the lift. Using a connection between a random $n$-lift of a graph and a randomly generated subgroup of the symmetric group on $n$-elements, we show that this probability is at least \u00a0$1 - O\\left(\\frac{1}{n^{\\gamma(\\delta)}}\\right)$ where $\\gamma(\\delta)&gt;0$ for $\\delta\\ge 5$ and it is strictly increasing with $\\delta$. We extend this to show that one may allow $\\delta$ to grow slowly as a function of the degree of the lift and the number of vertices and still obtain that random lifts are a.a.s. $\\delta$-connected. We also simplify a later result showing a lower bound on the edge expansion of random lifts. On a related note, we calculate the probability that a subgroup of a wreath product of symmetric groups generated by random generators is transitive, extending a well known result of Dixon which covers the case for subgroups of the symmetric group.<\/jats:p>","DOI":"10.37236\/6639","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T15:59:15Z","timestamp":1578671955000},"source":"Crossref","is-referenced-by-count":0,"title":["$\\delta$-Connectivity in Random Lifts of Graphs"],"prefix":"10.37236","volume":"24","author":[{"given":"Shashwat","family":"Silas","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2017,3,17]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v24i1p46\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v24i1p46\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T05:03:21Z","timestamp":1579237401000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v24i1p46"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,3,17]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2017,1,20]]}},"URL":"https:\/\/doi.org\/10.37236\/6639","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2017,3,17]]},"article-number":"P1.46"}}