{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,10,25]],"date-time":"2023-10-25T08:45:04Z","timestamp":1698223504201},"reference-count":6,"publisher":"Wiley","issue":"8","license":[{"start":{"date-parts":[[2006,10,6]],"date-time":"2006-10-06T00:00:00Z","timestamp":1160092800000},"content-version":"vor","delay-in-days":4327,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Graph Theory"],"published-print":{"date-parts":[[1994,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>One of the most fundamental results concerning paths in graphs is due to Ore: In a graph <jats:italic>G<\/jats:italic>, if deg <jats:italic>x<\/jats:italic> + deg <jats:italic>y<\/jats:italic> \u2267 |<jats:italic>V<\/jats:italic>(<jats:italic>G<\/jats:italic>)| + 1 for all pairs of nonadjacent vertices <jats:italic>x, y<\/jats:italic> \u2245 <jats:italic>V<\/jats:italic>(<jats:italic>G<\/jats:italic>), then <jats:italic>G<\/jats:italic> is hamiltonian\u2010connected. We generalize this result using set degrees. That is, for <jats:italic>S<\/jats:italic> \u2282 <jats:italic>V<\/jats:italic>(<jats:italic>G<\/jats:italic>), let deg <jats:italic>S<\/jats:italic> = |<jats:sub><jats:italic>x\u2245S<\/jats:italic><\/jats:sub> <jats:italic>N<\/jats:italic>(<jats:italic>x<\/jats:italic>)|, where <jats:italic>N<\/jats:italic>(<jats:italic>x<\/jats:italic>) = {<jats:italic>v<\/jats:italic>|<jats:italic>xv<\/jats:italic> \u2245 <jats:italic>E<\/jats:italic>(<jats:italic>G<\/jats:italic>)} is the neighborhood of <jats:italic>x<\/jats:italic>. In particular we show: In a 3\u2010connected graph <jats:italic>G<\/jats:italic>, if deg <jats:italic>S<\/jats:italic><jats:sub>1<\/jats:sub> + deg <jats:italic>S<\/jats:italic><jats:sub>2<\/jats:sub> \u2267 |<jats:italic>V<\/jats:italic>(<jats:italic>G<\/jats:italic>)| + 1 for each pair of distinct 2\u2010sets of vertices <jats:italic>S<\/jats:italic><jats:sub>1<\/jats:sub>, <jats:italic>S<\/jats:italic><jats:sub>2<\/jats:sub> \u2282 <jats:italic>V<\/jats:italic>(<jats:italic>G<\/jats:italic>), then <jats:italic>G<\/jats:italic> is hamiltonian\u2010connected.<\/jats:p><jats:p>Several corollaries and related results are also discussed.<\/jats:p>","DOI":"10.1002\/jgt.3190180808","type":"journal-article","created":{"date-parts":[[2007,5,26]],"date-time":"2007-05-26T13:46:42Z","timestamp":1180187202000},"page":"841-860","source":"Crossref","is-referenced-by-count":0,"title":["On hamiltonian\u2010connected graphs"],"prefix":"10.1002","volume":"18","author":[{"given":"Ronald J.","family":"Gould","sequence":"first","affiliation":[]},{"given":"Xingxing","family":"Yu","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2006,10,6]]},"reference":[{"key":"e_1_2_1_2_2","doi-asserted-by":"publisher","DOI":"10.1112\/plms\/s3-2.1.69"},{"key":"e_1_2_1_3_2","volume-title":"Graph Theory","author":"Gould R. J.","year":"1988"},{"key":"e_1_2_1_4_2","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365X(92)90132-Y"},{"key":"e_1_2_1_5_2","first-page":"93","article-title":"Generalized degrees, connectivity and Hamiltonian properties in graphs","volume":"16","author":"Faudree R. J.","year":"1991","journal-title":"J. Combinat. Compu. Inform. Sci."},{"key":"e_1_2_1_6_2","doi-asserted-by":"publisher","DOI":"10.2307\/2308928"},{"key":"e_1_2_1_7_2","first-page":"21","article-title":"Hamilton connected graphs","volume":"42","author":"Ore O.","year":"1963","journal-title":"J. Math. Pure. Appl."}],"container-title":["Journal of Graph Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fjgt.3190180808","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/jgt.3190180808","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,24]],"date-time":"2023-10-24T14:24:41Z","timestamp":1698157481000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/jgt.3190180808"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1994,12]]},"references-count":6,"journal-issue":{"issue":"8","published-print":{"date-parts":[[1994,12]]}},"alternative-id":["10.1002\/jgt.3190180808"],"URL":"https:\/\/doi.org\/10.1002\/jgt.3190180808","archive":["Portico"],"relation":{},"ISSN":["0364-9024","1097-0118"],"issn-type":[{"value":"0364-9024","type":"print"},{"value":"1097-0118","type":"electronic"}],"subject":[],"published":{"date-parts":[[1994,12]]}}}