{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:44:23Z","timestamp":1753893863878,"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>In this note we establish a resilience version of the classical hitting time result of Bollob\u00e1s and Thomason regarding connectivity. A graph $G$ is said to be $\\alpha$-resilient with respect to a monotone increasing graph property $\\mathcal{P}$ if for every spanning subgraph $H \\subseteq G$ satisfying $\\deg_H(v) \\leqslant \\alpha \\deg_G(v)$ for all $v \\in V(G)$, the graph $G - H$ still possesses $\\mathcal{P}$. Let $\\{G_i\\}$ be the random graph process, that is a process where, starting with an empty graph on $n$ vertices $G_0$, in each step $i \\geqslant 1$ an edge $e$ is chosen uniformly at random among the missing ones and added to the graph $G_{i - 1}$. We show that the random graph process is almost surely such that starting from $m \\geqslant (\\tfrac{1}{6} + o(1)) n \\log n$, the largest connected component of $G_m$ is $(\\tfrac{1}{2} - o(1))$-resilient with respect to connectivity. The result is optimal in the sense that the constants $1\/6$ in the number of edges and $1\/2$ in the resilience cannot be improved upon. We obtain similar results for $k$-connectivity.<\/jats:p>","DOI":"10.37236\/7885","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T02:14:18Z","timestamp":1578622458000},"source":"Crossref","is-referenced-by-count":0,"title":["On Resilience of Connectivity in the Evolution of Random Graphs"],"prefix":"10.37236","volume":"26","author":[{"given":"Luc","family":"Haller","sequence":"first","affiliation":[]},{"given":"Milo\u0161","family":"Truji\u0107","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2019,5,17]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v26i2p24\/7837","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v26i2p24\/7837","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,16]],"date-time":"2020-01-16T23:14:25Z","timestamp":1579216465000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v26i2p24"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,5,17]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2019,4,5]]}},"URL":"https:\/\/doi.org\/10.37236\/7885","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2019,5,17]]},"article-number":"P2.24"}}