{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,10,26]],"date-time":"2023-10-26T15:01:20Z","timestamp":1698332480410},"reference-count":10,"publisher":"Wiley","issue":"3","license":[{"start":{"date-parts":[[2006,10,5]],"date-time":"2006-10-05T00:00:00Z","timestamp":1160006400000},"content-version":"vor","delay-in-days":4844,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Graph Theory"],"published-print":{"date-parts":[[1993,7]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>For a graph <jats:italic>G<\/jats:italic> and an integer <jats:italic>k<\/jats:italic>, denote by <jats:italic>V<\/jats:italic><jats:sub><jats:italic>k<\/jats:italic><\/jats:sub> the set {<jats:italic>v<\/jats:italic> \u2208 <jats:italic>V<\/jats:italic>(<jats:italic>G<\/jats:italic>) | <jats:italic>d<\/jats:italic>(<jats:italic>v<\/jats:italic>) \u2265 <jats:italic>k<\/jats:italic>}. Veldman proved that if <jats:italic>G<\/jats:italic> is a 2\u2010connected graph of order <jats:italic>n<\/jats:italic> with <jats:italic>n<\/jats:italic> \u2264 <jats:italic>3k \u2010 2<\/jats:italic> and |<jats:italic>V<\/jats:italic><jats:sub><jats:italic>k<\/jats:italic><\/jats:sub>| \u2264 <jats:italic>k<\/jats:italic>, then <jats:italic>G<\/jats:italic> has a cycle containing all vertices of <jats:italic>V<\/jats:italic><jats:sub><jats:italic>k<\/jats:italic><\/jats:sub>. It is shown that the upper bound <jats:italic>k<\/jats:italic> on |<jats:italic>V<\/jats:italic><jats:sub><jats:italic>k<\/jats:italic><\/jats:sub>| is close to best possible in general. For the special case <jats:italic>k<\/jats:italic> = \u03b4(<jats:italic>G<\/jats:italic>), it is conjectured that the condition |<jats:italic>V<\/jats:italic><jats:sub><jats:italic>k<\/jats:italic><\/jats:sub>| \u2264 <jats:italic>k<\/jats:italic> can be omitted. Using a variation of Woodall's Hopping Lemma, the conjecture is proved under the additional condition that <jats:italic>n<\/jats:italic> \u2264 <jats:italic>2<\/jats:italic>\u03b4(<jats:italic>G<\/jats:italic>) + \u03b4(<jats:italic>G<\/jats:italic>) + <jats:italic>1<\/jats:italic>. This result is an almost\u2010generalization of Jackson's Theorem that every 2\u2010connected <jats:italic>k<\/jats:italic>\u2010regular graph of order <jats:italic>n<\/jats:italic> with <jats:italic>n<\/jats:italic> \u2264 <jats:italic>3k<\/jats:italic> is hamiltonian. An alternative proof of an extension of Jackson's Theorem is also presented. \u00a9 1993 John Wiley &amp; Sons, Inc.<\/jats:p>","DOI":"10.1002\/jgt.3190170311","type":"journal-article","created":{"date-parts":[[2007,5,26]],"date-time":"2007-05-26T13:18:25Z","timestamp":1180185505000},"page":"373-385","source":"Crossref","is-referenced-by-count":2,"title":["Cycles containing all vertices of maximum degree"],"prefix":"10.1002","volume":"17","author":[{"given":"H. J.","family":"Broersma","sequence":"first","affiliation":[]},{"given":"J. Den","family":"Van Heuvel","sequence":"additional","affiliation":[]},{"given":"H. A.","family":"Jung","sequence":"additional","affiliation":[]},{"given":"H. J.","family":"Veldman","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2006,10,5]]},"reference":[{"key":"e_1_2_1_2_2","unstructured":"B.Bollob\u00e1sandG.Brightwell Cycles through specified vertices. Preprint (1990)."},{"key":"e_1_2_1_3_2","doi-asserted-by":"publisher","DOI":"10.1016\/0095-8956(88)90086-X"},{"key":"e_1_2_1_4_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-349-03521-2"},{"key":"e_1_2_1_5_2","unstructured":"P.Fraisse D\u03bb\u2010cycles and their applications for hamiltonian graphs. Preprint (1986)."},{"key":"e_1_2_1_6_2","doi-asserted-by":"publisher","DOI":"10.1016\/0095-8956(80)90042-8"},{"key":"e_1_2_1_7_2","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.3190160310"},{"key":"e_1_2_1_8_2","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365X(83)90196-6"},{"key":"e_1_2_1_9_2","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365X(92)90612-J"},{"key":"e_1_2_1_10_2","doi-asserted-by":"publisher","DOI":"10.1016\/0095-8956(73)90038-5"},{"key":"e_1_2_1_11_2","first-page":"237","article-title":"An improvement of Jackson's result on Hamilton cycles in 2\u2010connected regular graphs","volume":"27","author":"Yongjin Zhu","year":"1985","journal-title":"Ann. Discrete Math."}],"container-title":["Journal of Graph Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fjgt.3190170311","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/jgt.3190170311","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,25]],"date-time":"2023-10-25T12:08:54Z","timestamp":1698235734000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/jgt.3190170311"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1993,7]]},"references-count":10,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1993,7]]}},"alternative-id":["10.1002\/jgt.3190170311"],"URL":"https:\/\/doi.org\/10.1002\/jgt.3190170311","archive":["Portico"],"relation":{},"ISSN":["0364-9024","1097-0118"],"issn-type":[{"value":"0364-9024","type":"print"},{"value":"1097-0118","type":"electronic"}],"subject":[],"published":{"date-parts":[[1993,7]]}}}