{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,28]],"date-time":"2025-10-28T00:05:48Z","timestamp":1761609948829,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>In 1962, P\u00f3sa conjectured that a graph $G=(V, E)$ contains a square of a Hamiltonian cycle if $\\delta(G)\\ge 2n\/3$. Only more than thirty years later Koml\u00f3s, S\u00e1rk\u0151zy, and Szemer\u00e9di proved this conjecture using the so-called Blow-Up Lemma. Here we extend their result to a random graph setting. We show that for every $\\epsilon &gt; 0$ and $p=n^{-1\/2+\\epsilon}$ a.a.s. every subgraph of $G_{n,p}$ with minimum degree at least $(2\/3+\\epsilon)np$ contains the square of a cycle on $(1-o(1))n$ vertices. This is almost best possible in three ways: (1) for $p\\ll n^{-1\/2}$ the random graph will not contain any square of a long cycle (2) one cannot hope for a resilience version for the square of a spanning cycle (as deleting all edges in the neighborhood of single vertex destroys this property) and (3) for $c&lt;2\/3$ a.a.s. $G_{n,p}$ contains a subgraph with minimum degree at least $cnp$ which does not contain the square of a path on $(1\/3+c)n$ vertices.<\/jats:p>","DOI":"10.37236\/6281","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T15:45:06Z","timestamp":1578671106000},"source":"Crossref","is-referenced-by-count":2,"title":["Local Resilience for Squares of Almost Spanning Cycles in Sparse Random Graphs"],"prefix":"10.37236","volume":"24","author":[{"given":"Andreas","family":"Noever","sequence":"first","affiliation":[]},{"given":"Angelika","family":"Steger","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2017,10,6]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v24i4p8\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v24i4p8\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T04:45:25Z","timestamp":1579236325000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v24i4p8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,10,6]]},"references-count":0,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2017,10,5]]}},"URL":"https:\/\/doi.org\/10.37236\/6281","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2017,10,6]]},"article-number":"P4.8"}}