{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,10,26]],"date-time":"2023-10-26T08:50:51Z","timestamp":1698310251102},"reference-count":7,"publisher":"Wiley","issue":"3","license":[{"start":{"date-parts":[[2006,10,3]],"date-time":"2006-10-03T00:00:00Z","timestamp":1159833600000},"content-version":"vor","delay-in-days":7337,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Graph Theory"],"published-print":{"date-parts":[[1986,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Clark proved that <jats:italic>L<\/jats:italic>(<jats:italic>G<\/jats:italic>) is hamiltonian if <jats:italic>G<\/jats:italic> is a connected graph of order <jats:italic>n<\/jats:italic> \u2265 6 such that deg <jats:italic>u<\/jats:italic> + deg <jats:italic>v<\/jats:italic> \u2265 <jats:italic>n<\/jats:italic> \u2013 1 \u2013 <jats:italic>p<\/jats:italic>(<jats:italic>n<\/jats:italic>) for every edge <jats:italic>uv<\/jats:italic> of <jats:italic>G<\/jats:italic>, where <jats:italic>p<\/jats:italic>(<jats:italic>n<\/jats:italic>) = 0 if <jats:italic>n<\/jats:italic> is even and <jats:italic>p<\/jats:italic>(<jats:italic>n<\/jats:italic>) = 1 if <jats:italic>n<\/jats:italic> is odd. Here it is shown that the bound <jats:italic>n<\/jats:italic> \u2013 1 \u2013 <jats:italic>p<\/jats:italic>(<jats:italic>n<\/jats:italic>) can be decreased to (2<jats:italic>n<\/jats:italic> + 1)\/3 if every bridge of <jats:italic>G<\/jats:italic> is incident with a vertex of degree 1, which is a necessary condition for hamiltonicity of <jats:italic>L<\/jats:italic>(<jats:italic>G<\/jats:italic>). Moreover, the conclusion that <jats:italic>L<\/jats:italic>(<jats:italic>G<\/jats:italic>) is hamiltonian can be strengthened to the conclusion that <jats:italic>L<\/jats:italic>(<jats:italic>G<\/jats:italic>) is pancyclic. Lesniak\u2010Foster and Williamson proved that <jats:italic>G<\/jats:italic> contains a spanning closed trail if |<jats:italic>V<\/jats:italic>(<jats:italic>G<\/jats:italic>)| = <jats:italic>n<\/jats:italic> \u2265 6, \u03b4(<jats:italic>G<\/jats:italic>) \u2265 2 and deg <jats:italic>u<\/jats:italic> + deg <jats:italic>v<\/jats:italic> \u2265 <jats:italic>n<\/jats:italic> \u2013 1 for every pair of nonadjacent vertices <jats:italic>u<\/jats:italic> and <jats:italic>v<\/jats:italic>. The bound <jats:italic>n<\/jats:italic> \u2013 1 can be decreased to (2<jats:italic>n<\/jats:italic> + 3)\/3 if <jats:italic>G<\/jats:italic> is connected and bridgeless, which is necessary for <jats:italic>G<\/jats:italic> to have a spanning closed trail.<\/jats:p>","DOI":"10.1002\/jgt.3190100317","type":"journal-article","created":{"date-parts":[[2007,6,9]],"date-time":"2007-06-09T07:24:09Z","timestamp":1181373849000},"page":"411-425","source":"Crossref","is-referenced-by-count":27,"title":["On circuits and pancyclic line graphs"],"prefix":"10.1002","volume":"10","author":[{"given":"A.","family":"Benhocine","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"L.","family":"Clark","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"N.","family":"K\u00f6hler","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"H. J.","family":"Veldman","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2006,10,3]]},"reference":[{"key":"e_1_2_1_2_2","unstructured":"A.Benhocine Probl\u00e8mes hamiltoniens et pancyclisme dans les graphes adjoints. Thesis Paris XI France (1986)."},{"key":"e_1_2_1_3_2","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.3190050312"},{"key":"e_1_2_1_4_2","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.3190080210"},{"key":"e_1_2_1_5_2","doi-asserted-by":"publisher","DOI":"10.21236\/AD0705364"},{"key":"e_1_2_1_6_2","doi-asserted-by":"publisher","DOI":"10.4153\/CMB-1965-051-3"},{"key":"e_1_2_1_7_2","doi-asserted-by":"publisher","DOI":"10.4153\/CMB-1977-034-8"},{"key":"e_1_2_1_8_2","article-title":"Existence of spanning and dominating trails and circuits","author":"Veldman H. J.","journal-title":"J. Graph Theory."}],"container-title":["Journal of Graph Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fjgt.3190100317","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/jgt.3190100317","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,21]],"date-time":"2023-10-21T18:46:39Z","timestamp":1697913999000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/jgt.3190100317"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1986,9]]},"references-count":7,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1986,9]]}},"alternative-id":["10.1002\/jgt.3190100317"],"URL":"https:\/\/doi.org\/10.1002\/jgt.3190100317","archive":["Portico"],"relation":{},"ISSN":["0364-9024","1097-0118"],"issn-type":[{"value":"0364-9024","type":"print"},{"value":"1097-0118","type":"electronic"}],"subject":[],"published":{"date-parts":[[1986,9]]}}}