{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,10,23]],"date-time":"2023-10-23T20:03:09Z","timestamp":1698091389165},"reference-count":8,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2006,10,5]],"date-time":"2006-10-05T00:00:00Z","timestamp":1160006400000},"content-version":"vor","delay-in-days":6792,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Graph Theory"],"published-print":{"date-parts":[[1988,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Let \ud835\udca2(<jats:italic>n, m<\/jats:italic>) denote the class of simple graphs on <jats:italic>n<\/jats:italic> vertices and <jats:italic>m<\/jats:italic> edges and let <jats:italic>G<\/jats:italic> \u2208 \ud835\udca2 (<jats:italic>n, m<\/jats:italic>). There are many results in graph theory giving conditions under which <jats:italic>G<\/jats:italic> contains certain types of subgraphs, such as cycles of given lengths, complete graphs, etc. For example, Turan's theorem gives a sufficient condition for <jats:italic>G<\/jats:italic> to contain a <jats:italic>K<\/jats:italic><jats:sub><jats:italic>k<\/jats:italic> + 1<\/jats:sub> in terms of the number of edges in <jats:italic>G<\/jats:italic>. In this paper we prove that, for <jats:italic>m<\/jats:italic> = \u03b1<jats:italic>n<\/jats:italic><jats:sup>2<\/jats:sup>, \u03b1 &gt; (<jats:italic>k<\/jats:italic> \u2010 1)\/2<jats:italic>k<\/jats:italic>, <jats:italic>G<\/jats:italic> contains a <jats:italic>K<\/jats:italic><jats:sub><jats:italic>k<\/jats:italic> + 1<\/jats:sub>, each vertex of which has degree at least <jats:italic>f<\/jats:italic>(\u03b1)<jats:italic>n<\/jats:italic> and determine the best possible <jats:italic>f<\/jats:italic>(\u03b1). For <jats:italic>m<\/jats:italic> = \u230a<jats:italic>n<\/jats:italic><jats:sup>2<\/jats:sup>\/4\u230b + 1 we establish that <jats:italic>G<\/jats:italic> contains cycles whose vertices have certain minimum degrees. Further, for <jats:italic>m<\/jats:italic> = \u03b1<jats:italic>n<\/jats:italic><jats:sup>2<\/jats:sup>, \u03b1 &gt; 0 we establish that <jats:italic>G<\/jats:italic> contains a subgraph <jats:italic>H<\/jats:italic> with \u03b4(<jats:italic>H<\/jats:italic>) \u2265 <jats:italic>f<\/jats:italic>(\u03b1, <jats:italic>n<\/jats:italic>) and determine the best possible value of <jats:italic>f<\/jats:italic>(\u03b1, <jats:italic>n<\/jats:italic>).<\/jats:p>","DOI":"10.1002\/jgt.3190120104","type":"journal-article","created":{"date-parts":[[2007,6,9]],"date-time":"2007-06-09T04:22:40Z","timestamp":1181362960000},"page":"17-27","source":"Crossref","is-referenced-by-count":5,"title":["Graphs with unavoidable subgraphs with large degrees"],"prefix":"10.1002","volume":"12","author":[{"given":"L.","family":"Caccetta","sequence":"first","affiliation":[]},{"given":"P.","family":"Erd\u00f6s","sequence":"additional","affiliation":[]},{"given":"K.","family":"Vijayan","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2006,10,5]]},"reference":[{"key":"e_1_2_1_2_2","volume-title":"Extremal Graph Theory","author":"Bollob\u00e1s B.","year":"1978"},{"key":"e_1_2_1_3_2","volume-title":"Graph Theory with Applications","author":"Bondy J. A.","year":"1977"},{"key":"e_1_2_1_4_2","first-page":"51","article-title":"On cycles in graphs","volume":"23","author":"Caccetta L.","year":"1987","journal-title":"Ars Combinat."},{"key":"e_1_2_1_5_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF02759702"},{"key":"e_1_2_1_6_2","first-page":"81","volume-title":"Proceedings of Southeastern Conference on Combinatorics, Graph Theory and Computing","author":"Erd\u00f6s P.","year":"1985"},{"key":"e_1_2_1_7_2","article-title":"Degree sum for a triangle in a graph","author":"Fan G.","journal-title":"J. Graph Theory"},{"key":"e_1_2_1_8_2","first-page":"436","article-title":"On an extremal problem in graph theory [in Hungarian]","volume":"48","author":"Tur\u00e1n P.","year":"1941","journal-title":"Mat. Fiz. Lapok"},{"key":"e_1_2_1_9_2","doi-asserted-by":"crossref","first-page":"19","DOI":"10.4064\/cm-3-1-19-30","article-title":"On the theory of graphs","volume":"3","author":"Tur\u00e1n P.","year":"1954","journal-title":"Colloq. Math."}],"container-title":["Journal of Graph Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fjgt.3190120104","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/jgt.3190120104","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,21]],"date-time":"2023-10-21T03:57:14Z","timestamp":1697860634000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/jgt.3190120104"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1988,3]]},"references-count":8,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1988,3]]}},"alternative-id":["10.1002\/jgt.3190120104"],"URL":"https:\/\/doi.org\/10.1002\/jgt.3190120104","archive":["Portico"],"relation":{},"ISSN":["0364-9024","1097-0118"],"issn-type":[{"value":"0364-9024","type":"print"},{"value":"1097-0118","type":"electronic"}],"subject":[],"published":{"date-parts":[[1988,3]]}}}