{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T10:41:56Z","timestamp":1740134516894,"version":"3.37.3"},"reference-count":10,"publisher":"Wiley","license":[{"start":{"date-parts":[[2018,1,1]],"date-time":"2018-01-01T00:00:00Z","timestamp":1514764800000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"MetACOnc","award":["ANR-15-CE40-0014"],"award-info":[{"award-number":["ANR-15-CE40-0014"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2018]]},"abstract":"Given a graph, its 2<\/mml:mn><\/mml:mrow><\/mml:math>-core<\/jats:italic> is the maximal subgraph of G<\/mml:mi><\/mml:mrow><\/mml:math> without vertices of degree 1<\/mml:mn><\/mml:mrow><\/mml:math>. A 2<\/mml:mn><\/mml:mrow><\/mml:math>-path in a connected graph is a simple path in its 2<\/mml:mn><\/mml:mrow><\/mml:math>-core such that all vertices in the path have degree 2<\/mml:mn><\/mml:mrow><\/mml:math>, except the endpoints which have degree \u2a7e<\/mml:mo>3<\/mml:mn><\/mml:math>. Consider the Erd\u0151s-R\u00e9nyi random graph G<\/mml:mi>(<\/mml:mo>n<\/mml:mi>,<\/mml:mo>M<\/mml:mi>)<\/mml:mo><\/mml:math> built with n<\/mml:mi><\/mml:mrow><\/mml:math> vertices and M<\/mml:mi><\/mml:mrow><\/mml:math> edges uniformly randomly chosen from the set of n<\/mml:mi><\/mml:mtd><\/mml:mtr>2<\/mml:mn><\/mml:mtd><\/mml:mtr><\/mml:mtable><\/mml:mrow><\/mml:mfenced><\/mml:mrow><\/mml:math> edges. Let \u03be<\/mml:mi><\/mml:mrow>n<\/mml:mi>,<\/mml:mo>M<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math> be the maximum 2<\/mml:mn><\/mml:mrow><\/mml:math>-path length of G<\/mml:mi>(<\/mml:mo>n<\/mml:mi>,<\/mml:mo>M<\/mml:mi>)<\/mml:mo><\/mml:math>. In this paper, we determine that there exists a constant c<\/mml:mi>(<\/mml:mo>\u03bb<\/mml:mi>)<\/mml:mo><\/mml:math> such that E<\/mml:mi>\u03be<\/mml:mi><\/mml:mrow>n<\/mml:mi>,<\/mml:mo>n<\/mml:mi><\/mml:mrow>\/<\/mml:mo>2<\/mml:mn><\/mml:mrow><\/mml:mrow><\/mml:mrow><\/mml:mfenced>1<\/mml:mn>+<\/mml:mo>\u03bb<\/mml:mi>n<\/mml:mi><\/mml:mrow>-<\/mml:mo>1<\/mml:mn>\/<\/mml:mo>3<\/mml:mn><\/mml:mrow><\/mml:msup><\/mml:mrow><\/mml:mfenced><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:mfenced>~<\/mml:mo>c<\/mml:mi>(<\/mml:mo>\u03bb<\/mml:mi>)<\/mml:mo>n<\/mml:mi><\/mml:mrow>1<\/mml:mn>\/<\/mml:mo>3<\/mml:mn><\/mml:mrow><\/mml:msup>,<\/mml:mo>\u2009<\/mml:mo>\u2009<\/mml:mo>f<\/mml:mi>o<\/mml:mi>r<\/mml:mi>\u2009<\/mml:mo>\u2009<\/mml:mo>a<\/mml:mi>n<\/mml:mi>y<\/mml:mi>\u2009<\/mml:mo>\u2009<\/mml:mo>r<\/mml:mi>e<\/mml:mi>a<\/mml:mi>l<\/mml:mi>\u2009<\/mml:mo>\u2009<\/mml:mo>\u03bb<\/mml:mi>.<\/mml:mo><\/mml:math> This parameter is studied through the use of generating functions and complex analysis.<\/jats:p>","DOI":"10.1155\/2018\/8983218","type":"journal-article","created":{"date-parts":[[2018,5,14]],"date-time":"2018-05-14T19:32:39Z","timestamp":1526326359000},"page":"1-5","source":"Crossref","is-referenced-by-count":0,"title":["The Maximal Length of 2-Path in Random Critical Graphs"],"prefix":"10.1155","volume":"2018","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8208-4735","authenticated-orcid":true,"given":"Vonjy","family":"Rasendrahasina","sequence":"first","affiliation":[{"name":"ENS-Universit\u00e9 d\u2019Antananarivo, Antananarivo, Madagascar"}]},{"given":"Vlady","family":"Ravelomanana","sequence":"additional","affiliation":[{"name":"IRIF UMR CNRS 8243, Universit\u00e9 Denis Diderot, Paris, France"}]},{"given":"Liva","family":"Aly Raonenantsoamihaja","sequence":"additional","affiliation":[{"name":"Facult\u00e9 des Sciences, Universit\u00e9 d\u2019Antananarivo, Antananarivo, Madagascar"}]}],"member":"311","reference":[{"key":"11","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.3240040303"},{"key":"5","first-page":"17","volume":"5","year":"1960","journal-title":"Publications of the Mathematical Institute of the Hungarian Academy of Sciences"},{"key":"6","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365X(89)90087-3"},{"key":"4","doi-asserted-by":"publisher","DOI":"10.1016\/j.ejc.2013.06.004"},{"key":"2","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.20342"},{"year":"1973","key":"9"},{"key":"1","first-page":"376","volume":"23","year":"1889","journal-title":"The Quarterly journal of pure and applied mathematics"},{"key":"13","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.3190010407"},{"key":"10","doi-asserted-by":"publisher","DOI":"10.1214\/07-PS104"},{"key":"15","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.3190040409"}],"container-title":["Journal of Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2018\/8983218.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2018\/8983218.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2018\/8983218.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2018,5,14]],"date-time":"2018-05-14T19:32:44Z","timestamp":1526326364000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.hindawi.com\/journals\/jam\/2018\/8983218\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018]]},"references-count":10,"alternative-id":["8983218","8983218"],"URL":"https:\/\/doi.org\/10.1155\/2018\/8983218","relation":{},"ISSN":["1110-757X","1687-0042"],"issn-type":[{"type":"print","value":"1110-757X"},{"type":"electronic","value":"1687-0042"}],"subject":[],"published":{"date-parts":[[2018]]}}}