{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,10]],"date-time":"2026-03-10T03:18:33Z","timestamp":1773112713125,"version":"3.50.1"},"reference-count":23,"publisher":"EDP Sciences","issue":"4","license":[{"start":{"date-parts":[[2022,8,24]],"date-time":"2022-08-24T00:00:00Z","timestamp":1661299200000},"content-version":"vor","delay-in-days":54,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11871239, 11971196,11871280,U1811461"],"award-info":[{"award-number":["11871239, 11971196,11871280,U1811461"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["RAIRO-Oper. Res."],"accepted":{"date-parts":[[2022,6,24]]},"published-print":{"date-parts":[[2022,7]]},"abstract":"<jats:p>For a graph <jats:italic>G<\/jats:italic>, a spanning subgraph <jats:italic>F<\/jats:italic> of <jats:italic>G<\/jats:italic> is called an {<jats:italic>P<\/jats:italic><jats:sub>2<\/jats:sub>, <jats:italic>P<\/jats:italic><jats:sub>5<\/jats:sub>}-factor if every component of <jats:italic>F<\/jats:italic> is isomorphic to <jats:italic>P<\/jats:italic><jats:sub>2<\/jats:sub> or <jats:italic>P<\/jats:italic><jats:sub>5<\/jats:sub>, where <jats:italic>P<\/jats:italic><jats:sub><jats:italic>k<\/jats:italic><\/jats:sub> denotes the path of order <jats:italic>k<\/jats:italic>. It was proved by Egawa and Furuya that if <jats:italic>G<\/jats:italic> satisfies 3<jats:italic>c<\/jats:italic><jats:sub>1<\/jats:sub> (<jats:italic>G<\/jats:italic>\u00a0\u2212\u00a0<jats:italic>S<\/jats:italic>) + 2<jats:italic>c<\/jats:italic><jats:sub>3<\/jats:sub> (<jats:italic>G<\/jats:italic>\u00a0\u2212\u00a0<jats:italic>S<\/jats:italic>) \u2264 4|<jats:italic>S<\/jats:italic>| + 1 for all <jats:italic>S<\/jats:italic> \u2286 <jats:italic>V<\/jats:italic>(<jats:italic>G<\/jats:italic>), then <jats:italic>G<\/jats:italic> has a {<jats:italic>P<\/jats:italic><jats:sub>2<\/jats:sub>, <jats:italic>P<\/jats:italic><jats:sub>5<\/jats:sub>}-factor, where <jats:italic>c<\/jats:italic><jats:sub><jats:italic>k<\/jats:italic><\/jats:sub> (<jats:italic>G<\/jats:italic>\u00a0\u2212\u00a0<jats:italic>S<\/jats:italic>) denotes the number of components of <jats:italic>G<\/jats:italic>\u00a0\u2212\u00a0<jats:italic>S<\/jats:italic> with order <jats:italic>k<\/jats:italic>. By this result, we give some other sufficient conditions for a graph to have a {<jats:italic>P<\/jats:italic><jats:sub>2<\/jats:sub>, <jats:italic>P<\/jats:italic><jats:sub>5<\/jats:sub>}-factor by various graphic parameters such as toughness, binding number, degree sums, etc. Moreover, we obtain some regular graphs and some <jats:italic>K<\/jats:italic><jats:sub>(1,<jats:italic>r<\/jats:italic>)<\/jats:sub>-free graphs having {<jats:italic>P<\/jats:italic><jats:sub>2<\/jats:sub>,\u00a0<jats:italic>P<\/jats:italic><jats:sub>5<\/jats:sub>}-factors.<\/jats:p>","DOI":"10.1051\/ro\/2022112","type":"journal-article","created":{"date-parts":[[2022,6,25]],"date-time":"2022-06-25T19:00:16Z","timestamp":1656183616000},"page":"2895-2901","source":"Crossref","is-referenced-by-count":12,"title":["Sufficient conditions for graphs with {<i>P<\/i><sub>2<\/sub>, <i>P<\/i><sub>5<\/sub>}-factors"],"prefix":"10.1051","volume":"56","author":[{"given":"Guowei","family":"Dai","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yicheng","family":"Hang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiaoyan","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zan-Bo","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2153-7339","authenticated-orcid":false,"given":"Wenqi","family":"Wang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"250","published-online":{"date-parts":[[2022,8,24]]},"reference":[{"key":"R1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1002\/jgt.3190090103","volume":"9","author":"Akiyama","year":"1985","journal-title":"J. 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