{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,10,24]],"date-time":"2023-10-24T11:14:30Z","timestamp":1698146070348},"reference-count":7,"publisher":"Wiley","issue":"4","license":[{"start":{"date-parts":[[2006,10,5]],"date-time":"2006-10-05T00:00:00Z","timestamp":1160006400000},"content-version":"vor","delay-in-days":5513,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Graph Theory"],"published-print":{"date-parts":[[1991,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Let <jats:italic>m<\/jats:italic> and <jats:italic>n<\/jats:italic> be nonnegative integers. Denote by <jats:italic>P<\/jats:italic>(<jats:italic>m,n<\/jats:italic>) the set of all triangle\u2010free graphs <jats:italic>G<\/jats:italic> such that for any independent <jats:italic>m<\/jats:italic>\u2010subset <jats:italic>M<\/jats:italic> and any <jats:italic>n<\/jats:italic>\u2010subset <jats:italic>N<\/jats:italic> of <jats:italic>V<\/jats:italic>(<jats:italic>G<\/jats:italic>) with <jats:italic>M<\/jats:italic> \u2229 <jats:italic>N<\/jats:italic> = \u00d8, there exists a unique vertex of <jats:italic>G<\/jats:italic> that is adjacent to each vertex in <jats:italic>M<\/jats:italic> and nonadjacent to any vertex in <jats:italic>N<\/jats:italic>. We prove that if <jats:italic>m<\/jats:italic> \u2a7e 2 and <jats:italic>n<\/jats:italic> \u2a7e 1, then <jats:italic>P<\/jats:italic>(<jats:italic>m,n<\/jats:italic>) = \u00d8 whenever <jats:italic>m<\/jats:italic> \u2a7d <jats:italic>n<\/jats:italic>, and <jats:italic>P<\/jats:italic>(<jats:italic>m,n<\/jats:italic>) = {<jats:italic>K<\/jats:italic><jats:sub><jats:italic>m,n<\/jats:italic>+<jats:italic>1<\/jats:italic><\/jats:sub>} whenever <jats:italic>m<\/jats:italic> &gt; <jats:italic>n<\/jats:italic>. We also have <jats:italic>P<\/jats:italic>(1,1) = {<jats:italic>C<\/jats:italic><jats:sub>5<\/jats:sub>} and <jats:italic>P<\/jats:italic>(1,<jats:italic>n<\/jats:italic>) = \u00d8 for <jats:italic>n<\/jats:italic> \u2a7e 2. In the degenerate cases, the class <jats:italic>P<\/jats:italic>(0,<jats:italic>n<\/jats:italic>) is completely determined, whereas the class <jats:italic>P<\/jats:italic>(<jats:italic>m<\/jats:italic>,0), which is most interesting, being rich in graphs, is partially determined.<\/jats:p>","DOI":"10.1002\/jgt.3190150404","type":"journal-article","created":{"date-parts":[[2007,6,8]],"date-time":"2007-06-08T09:27:58Z","timestamp":1181294878000},"page":"375-388","source":"Crossref","is-referenced-by-count":4,"title":["Characterization of a class of triangle\u2010free graphs with a certain adjacency property"],"prefix":"10.1002","volume":"15","author":[{"given":"Brian","family":"Alspach","sequence":"first","affiliation":[]},{"given":"C. C.","family":"Chen","sequence":"additional","affiliation":[]},{"given":"Katherine","family":"Heinrich","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2006,10,5]]},"reference":[{"key":"e_1_2_1_2_2","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511608704"},{"key":"e_1_2_1_3_2","doi-asserted-by":"publisher","DOI":"10.1016\/0095-8956(71)90031-1"},{"key":"e_1_2_1_4_2","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.3190030305"},{"key":"e_1_2_1_5_2","first-page":"287","article-title":"A property of random graphs","volume":"19","author":"Caccetta L.","year":"1985","journal-title":"Ars Combinat."},{"key":"e_1_2_1_6_2","first-page":"93","article-title":"On strongly accessible graphs","volume":"17","author":"Caccetta L.","year":"1984","journal-title":"Ars Combinat."},{"key":"e_1_2_1_7_2","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.3190050406"},{"key":"e_1_2_1_8_2","doi-asserted-by":"publisher","DOI":"10.1147\/rd.45.0497"}],"container-title":["Journal of Graph Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fjgt.3190150404","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/jgt.3190150404","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,23]],"date-time":"2023-10-23T03:46:41Z","timestamp":1698032801000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/jgt.3190150404"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1991,9]]},"references-count":7,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1991,9]]}},"alternative-id":["10.1002\/jgt.3190150404"],"URL":"https:\/\/doi.org\/10.1002\/jgt.3190150404","archive":["Portico"],"relation":{},"ISSN":["0364-9024","1097-0118"],"issn-type":[{"value":"0364-9024","type":"print"},{"value":"1097-0118","type":"electronic"}],"subject":[],"published":{"date-parts":[[1991,9]]}}}