{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,10,30]],"date-time":"2023-10-30T19:04:30Z","timestamp":1698692670100},"reference-count":11,"publisher":"Wiley","issue":"2","license":[{"start":{"date-parts":[[2006,10,11]],"date-time":"2006-10-11T00:00:00Z","timestamp":1160524800000},"content-version":"vor","delay-in-days":5031,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Random Struct Algorithms"],"published-print":{"date-parts":[[1993,1]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Let <jats:italic>G<jats:sub>n<\/jats:sub><\/jats:italic> = (<jats:italic>V<jats:sub>n<\/jats:sub>, E<jats:sub>n<\/jats:sub><\/jats:italic>) be the random subgraph of the <jats:italic>n<\/jats:italic>\u2010cube graph <jats:italic>Q<jats:sup>n<\/jats:sup><\/jats:italic> produced as follows: <jats:italic>V<jats:sub>n<\/jats:sub><\/jats:italic> is randomly sampled from the vertex set of <jats:italic>Q<jats:sup>n<\/jats:sup><\/jats:italic> so that <jats:italic>E<\/jats:italic>{<jats:italic>x<\/jats:italic> \u2208<jats:italic>V<jats:sub>n<\/jats:sub><\/jats:italic>} = <jats:italic>p<jats:sub>v<\/jats:sub><\/jats:italic> independently for each vertex <jats:italic>x<\/jats:italic> \u2208<jats:italic>Q<jats:sup>n<\/jats:sup><\/jats:italic>. Then <jats:italic>E<jats:sub>n<\/jats:sub><\/jats:italic> is randomly sampled from the set of edges induced by <jats:italic>V<jats:sub>n<\/jats:sub><\/jats:italic> in <jats:italic>Q<jats:sup>n<\/jats:sup><\/jats:italic> so that <jats:italic>P<\/jats:italic>{(<jats:italic>x, y<\/jats:italic>) \u2208 <jats:italic>E<jats:sub>n<\/jats:sub><\/jats:italic>} = <jats:italic>p<jats:sub>e<\/jats:sub><\/jats:italic> independently for each induced edge (<jats:italic>x, y<\/jats:italic>). Let <jats:italic>P<jats:sub>v<\/jats:sub><\/jats:italic> and <jats:italic>p<jats:sub>e<\/jats:sub><\/jats:italic> be fixed probabilities such that 0 &lt; <jats:italic>p<\/jats:italic> = <jats:italic>p<jats:sub>v<\/jats:sub>p<jats:sub>e<\/jats:sub><\/jats:italic> \u2a7d 1 and <jats:italic>m<jats:sub>p<\/jats:sub><\/jats:italic> = [\u22121 \/log<jats:sub>2<\/jats:sub>(1 \u2212 <jats:italic>p<\/jats:italic>)]. We show that for the radius <jats:italic>R(G<jats:sub>n<\/jats:sub>)<\/jats:italic> and the diameter <jats:italic>D(G<jats:sub>n<\/jats:sub>)<\/jats:italic> of the main component of <jats:italic>G<jats:sub>n<\/jats:sub><\/jats:italic> almost surely the following inequalities hold <jats:chem-struct-wrap><jats:chem-struct><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mimetype=\"image\/gif\" position=\"anchor\" specific-use=\"enlarged-web-image\" xlink:href=\"graphic\/must001.gif\"><jats:alt-text>magnified image<\/jats:alt-text><\/jats:graphic><\/jats:chem-struct><\/jats:chem-struct-wrap>. \u00a9 1993 John Wiley &amp; Sons, Inc.<\/jats:p>","DOI":"10.1002\/rsa.3240040207","type":"journal-article","created":{"date-parts":[[2007,6,1]],"date-time":"2007-06-01T19:56:32Z","timestamp":1180727792000},"page":"215-229","source":"Crossref","is-referenced-by-count":4,"title":["Radius and diameter of random subgraphs of the hypercube"],"prefix":"10.1002","volume":"4","author":[{"given":"A. V.","family":"Kostochka","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"A. A.","family":"Sapozhenko","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"K.","family":"Weber","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2006,10,11]]},"reference":[{"key":"e_1_2_1_2_2","first-page":"17","article-title":"On the strength of connectivity of random subgraphs of the","volume":"33","author":"Dyer M.","year":"1987","journal-title":"n\u2010cube. Ann. Discrete Math."},{"key":"e_1_2_1_3_2","volume-title":"An Introduction to Probability Theory and Its Applications","author":"Feller W.","year":"1959"},{"key":"e_1_2_1_4_2","unstructured":"U.KoniecznaandK.Weber Graph theoretic parameters of random subgraphs of then\u2010cube Dresdner Reiher zur Forschung 9\/88 Proc. Graphentheorie und ihre Anwendungen Stadt Wehlen May 16\u201320 1988 pp.25\u201328."},{"key":"e_1_2_1_5_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-46908-4_57"},{"key":"e_1_2_1_6_2","first-page":"91","article-title":"Metric properties of almost all Boolean funcitions","volume":"10","author":"Sapozhenko A. A.","year":"1967","journal-title":"Discretny Analiz."},{"key":"e_1_2_1_7_2","first-page":"227","article-title":"Geometric structure of almost all Boolean functions","volume":"30","author":"Sapozhenko A. A.","year":"1975","journal-title":"Probl. Kibern."},{"key":"e_1_2_1_8_2","first-page":"42","article-title":"On the number of connected subsets with given cardinality of the boundary in bipartite graphs","volume":"45","author":"Sapozhenko A. A.","year":"1987","journal-title":"Discretny Analiz."},{"key":"e_1_2_1_9_2","first-page":"507","volume-title":"Contemporary Methods in Graph Theory","author":"Sapozhenko A. A.","year":"1990"},{"key":"e_1_2_1_10_2","first-page":"111","article-title":"Geometric structure of random Boolean functions","volume":"35","author":"Toman E.","year":"1979","journal-title":"Probl. Kibern."},{"key":"e_1_2_1_11_2","first-page":"601","article-title":"On the components of random graphs in the n\u2010cube","volume":"22","author":"Weber K.","year":"1986","journal-title":"Electron. Inf. verarb. Kybern. EIK"},{"key":"e_1_2_1_12_2","unstructured":"K.Weber On the components of random subgraphs of then\u2010cube(submitted)."}],"container-title":["Random Structures &amp; Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Frsa.3240040207","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/rsa.3240040207","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,16]],"date-time":"2023-10-16T10:59:12Z","timestamp":1697453952000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/rsa.3240040207"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1993,1]]},"references-count":11,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1993,1]]}},"alternative-id":["10.1002\/rsa.3240040207"],"URL":"https:\/\/doi.org\/10.1002\/rsa.3240040207","archive":["Portico"],"relation":{},"ISSN":["1042-9832","1098-2418"],"issn-type":[{"value":"1042-9832","type":"print"},{"value":"1098-2418","type":"electronic"}],"subject":[],"published":{"date-parts":[[1993,1]]}}}