{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,28]],"date-time":"2025-09-28T12:48:09Z","timestamp":1759063689897},"reference-count":29,"publisher":"EDP Sciences","issue":"3","license":[{"start":{"date-parts":[[2021,5,19]],"date-time":"2021-05-19T00:00:00Z","timestamp":1621382400000},"content-version":"vor","delay-in-days":18,"URL":"https:\/\/www.edpsciences.org\/en\/authors\/copyright-and-licensing"}],"funder":[{"name":"This work is supported by Six Talent Peaks Project in Jiangsu Province, China","award":["Grant No. JY-022"],"award-info":[{"award-number":["Grant No. JY-022"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["RAIRO-Oper. Res."],"accepted":{"date-parts":[[2021,4,11]]},"published-print":{"date-parts":[[2021,5]]},"abstract":"<jats:p>A<jats:italic>P<\/jats:italic><jats:sub>\u2265<jats:italic>k<\/jats:italic><\/jats:sub>-factor of a graph<jats:italic>G<\/jats:italic>is a spanning subgraph of<jats:italic>G<\/jats:italic>whose components are paths of order at least<jats:italic>k<\/jats:italic>. We say that a graph<jats:italic>G<\/jats:italic>is<jats:italic>P<\/jats:italic><jats:sub>\u2265<jats:italic>k<\/jats:italic><\/jats:sub>-factor covered if for every edge<jats:italic>e<\/jats:italic>\u2208<jats:italic>E<\/jats:italic>(<jats:italic>G<\/jats:italic>),<jats:italic>G<\/jats:italic>admits a<jats:italic>P<\/jats:italic><jats:sub>\u2265<jats:italic>k<\/jats:italic><\/jats:sub>-factor that contains<jats:italic>e<\/jats:italic>; and we say that a graph<jats:italic>G<\/jats:italic>is<jats:italic>P<\/jats:italic><jats:sub>\u2265<jats:italic>k<\/jats:italic><\/jats:sub>-factor uniform if for every edge<jats:italic>e<\/jats:italic>\u2208<jats:italic>E<\/jats:italic>(<jats:italic>G<\/jats:italic>), the graph<jats:italic>G<\/jats:italic>\u2212<jats:italic>e<\/jats:italic>is<jats:italic>P<\/jats:italic><jats:sub>\u2265<jats:italic>k<\/jats:italic><\/jats:sub>-factor covered. In other words,<jats:italic>G<\/jats:italic>is<jats:italic>P<\/jats:italic><jats:sub>\u2265<jats:italic>k<\/jats:italic><\/jats:sub>-factor uniform if for every pair of edges<jats:italic>e<\/jats:italic><jats:sub>1<\/jats:sub>,<jats:italic>e<\/jats:italic><jats:sub>2<\/jats:sub>\u2208<jats:italic>E<\/jats:italic>(<jats:italic>G<\/jats:italic>),<jats:italic>G<\/jats:italic>admits a<jats:italic>P<\/jats:italic><jats:sub>\u2265<jats:italic>k<\/jats:italic><\/jats:sub>-factor that contains<jats:italic>e<\/jats:italic><jats:sub>1<\/jats:sub>and avoids<jats:italic>e<\/jats:italic><jats:sub>2<\/jats:sub>. In this article, we testify that (1) a 3-edge-connected graph<jats:italic>G<\/jats:italic>is<jats:italic>P<\/jats:italic><jats:sub>\u2265<jats:italic>k<\/jats:italic><\/jats:sub>-factor uniform if its isolated toughness<jats:italic>I<\/jats:italic>(<jats:italic>G<\/jats:italic>) &gt; 1; (2) a 3-edge-connected graph<jats:italic>G<\/jats:italic>is<jats:italic>P<\/jats:italic><jats:sub>\u2265<jats:italic>k<\/jats:italic><\/jats:sub>-factor uniform if its isolated toughness<jats:italic>I<\/jats:italic>(<jats:italic>G<\/jats:italic>) &gt; 2. Furthermore, we explain that these conditions on isolated toughness and edge-connectivity in our main results are best possible in some sense.<\/jats:p>","DOI":"10.1051\/ro\/2021061","type":"journal-article","created":{"date-parts":[[2021,4,13]],"date-time":"2021-04-13T19:04:13Z","timestamp":1618340653000},"page":"1279-1290","source":"Crossref","is-referenced-by-count":14,"title":["Isolated toughness and path-factor uniform graphs"],"prefix":"10.1051","volume":"55","author":[{"given":"Sizhong","family":"Zhou","sequence":"first","affiliation":[]},{"given":"Zhiren","family":"Sun","sequence":"additional","affiliation":[]},{"given":"Hongxia","family":"Liu","sequence":"additional","affiliation":[]}],"member":"250","published-online":{"date-parts":[[2021,5,19]]},"reference":[{"key":"R1","doi-asserted-by":"crossref","first-page":"983","DOI":"10.1002\/int.22233","volume":"35","author":"Abdullah","year":"2020","journal-title":"Int. J. Intell. 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