{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,14]],"date-time":"2026-05-14T21:49:27Z","timestamp":1778795367312,"version":"3.51.4"},"reference-count":5,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2006,10,3]],"date-time":"2006-10-03T00:00:00Z","timestamp":1159833600000},"content-version":"vor","delay-in-days":7886,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Graph Theory"],"published-print":{"date-parts":[[1985,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>A connected graph <jats:italic>G<\/jats:italic> is called <jats:italic>t<\/jats:italic>\u2010tough if <jats:italic>t<\/jats:italic> \u00b7 <jats:italic>w(G \u2010 S) \u2a7d |S|<\/jats:italic> for any subset <jats:italic>S<\/jats:italic> of <jats:italic>V(G)<\/jats:italic> with <jats:italic>w(G \u2010 S)<\/jats:italic> &gt; 1, where <jats:italic>w(G<\/jats:italic> \u2010 <jats:italic>S<\/jats:italic>) is the number of connected components of <jats:italic>G<\/jats:italic> \u2010 <jats:italic>S.<\/jats:italic> We prove that every <jats:italic>k<\/jats:italic>\u2010tough graph has a <jats:italic>k<\/jats:italic>\u2010factor if <jats:italic>k|G|<\/jats:italic> is even and <jats:italic>|G|<\/jats:italic> \u2a7e <jats:italic>k<\/jats:italic> + 1. This result, first conjectured by Chv\u00e1tal, is sharp in the following sense: For any positive integer <jats:italic>k<\/jats:italic> and for any positive real number \u03b5, there exists <jats:italic>a (k<\/jats:italic> \u2010 \u03b5)\u2010tough graph <jats:italic>G<\/jats:italic> with <jats:italic>k|G|<\/jats:italic> even and <jats:italic>|G|<\/jats:italic> \u2a7e <jats:italic>k<\/jats:italic> + 1 which has no <jats:italic>k<\/jats:italic>\u2010factor.<\/jats:p>","DOI":"10.1002\/jgt.3190090106","type":"journal-article","created":{"date-parts":[[2007,5,29]],"date-time":"2007-05-29T07:49:42Z","timestamp":1180424982000},"page":"87-95","source":"Crossref","is-referenced-by-count":121,"title":["Toughness and the existence of <i>k<\/i>\u2010factors"],"prefix":"10.1002","volume":"9","author":[{"given":"Hikoe","family":"Enomoto","sequence":"first","affiliation":[]},{"given":"Bill","family":"Jackson","sequence":"additional","affiliation":[]},{"given":"P.","family":"Katerinis","sequence":"additional","affiliation":[]},{"given":"Akira","family":"Saito","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2006,10,3]]},"reference":[{"key":"e_1_2_1_2_2","volume-title":"Graphs & Digraphs","author":"Behzad M.","year":"1979"},{"key":"e_1_2_1_3_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-349-03521-2"},{"key":"e_1_2_1_4_2","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365X(73)90138-6"},{"key":"e_1_2_1_5_2","unstructured":"N.Tsikopoulos Nearlyk\u2010tough graphs with nok\u2010factor unpublished."},{"key":"e_1_2_1_6_2","doi-asserted-by":"publisher","DOI":"10.4153\/CJM-1952-028-2"}],"container-title":["Journal of Graph Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fjgt.3190090106","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/jgt.3190090106","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,21]],"date-time":"2023-10-21T05:46:26Z","timestamp":1697867186000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/jgt.3190090106"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1985,3]]},"references-count":5,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1985,3]]}},"alternative-id":["10.1002\/jgt.3190090106"],"URL":"https:\/\/doi.org\/10.1002\/jgt.3190090106","archive":["Portico"],"relation":{},"ISSN":["0364-9024","1097-0118"],"issn-type":[{"value":"0364-9024","type":"print"},{"value":"1097-0118","type":"electronic"}],"subject":[],"published":{"date-parts":[[1985,3]]}}}