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Res."],"accepted":{"date-parts":[[2023,3,29]]},"published-print":{"date-parts":[[2023,3]]},"abstract":"<jats:p>An <jats:italic>\u2131<\/jats:italic>-factor is a spanning subgraph <jats:italic>H<\/jats:italic> such that each connected component of <jats:italic>H<\/jats:italic> is isomorphic to some graph in <jats:italic>\u2131<\/jats:italic>. We use <jats:italic>P<\/jats:italic><jats:sub><jats:italic>k<\/jats:italic><\/jats:sub> and <jats:italic>K<\/jats:italic><jats:sub>1,<jats:italic>r<\/jats:italic><\/jats:sub> to denote the path of order <jats:italic>k<\/jats:italic> and the star of order <jats:italic>r<\/jats:italic>\u00a0+\u00a01, respectively. In particular, <jats:italic>H<\/jats:italic> is called a {<jats:italic>P<\/jats:italic><jats:sub>2<\/jats:sub>,\u00a0<jats:italic>P<\/jats:italic><jats:sub>3<\/jats:sub>}-factor of <jats:italic>G<\/jats:italic> if <jats:italic>\u2131<\/jats:italic> = {<jats:italic>P<jats:sub>2<\/jats:sub><\/jats:italic>, <jats:italic>P<jats:sub>3<\/jats:sub><\/jats:italic>}; <jats:italic>H<\/jats:italic> is called a <jats:italic>P<\/jats:italic>\u2265<jats:italic><jats:sub>k<\/jats:sub><\/jats:italic>-factor of <jats:italic>G<\/jats:italic> if <jats:italic>\u2131<\/jats:italic> = {<jats:italic>P<jats:sub>k<\/jats:sub><\/jats:italic>, <jats:italic>P<jats:sub>k+1<\/jats:sub><\/jats:italic>,\u2026}, where <jats:italic>k<\/jats:italic> \u2265 2; <jats:italic>H<\/jats:italic> is called an <jats:italic>S<jats:sub>n<\/jats:sub><\/jats:italic>-factor of <jats:italic>G<\/jats:italic> if <jats:italic>\u2131<\/jats:italic> = {<jats:italic>P<jats:sub>2<\/jats:sub><\/jats:italic>, <jats:italic>P<jats:sub>3<\/jats:sub><\/jats:italic>, <jats:italic>K<\/jats:italic><jats:sub>1,3<\/jats:sub>,\u2026, <jats:italic>K<jats:sub>1,<jats:italic>n<\/jats:italic><\/jats:sub><\/jats:italic>}, where <jats:italic>n<\/jats:italic> \u2265 2. A graph <jats:italic>G<\/jats:italic> is called a <jats:italic>\u2131<\/jats:italic>\u2265<jats:sub><jats:italic>k<\/jats:italic><\/jats:sub>-factor covered graph if there is a <jats:italic>\u2131<\/jats:italic>\u2265<jats:sub><jats:italic>k<\/jats:italic><\/jats:sub>-factor of <jats:italic>G<\/jats:italic> including <jats:italic>e<\/jats:italic> for any <jats:italic>e<\/jats:italic> \u2208 <jats:italic>E<\/jats:italic>(<jats:italic>G<\/jats:italic>). We call a graph <jats:italic>G<\/jats:italic> is <jats:italic>K<\/jats:italic><jats:sub>1,<jats:italic>r<\/jats:italic><\/jats:sub>-free if <jats:italic>G<\/jats:italic> does not contain an induced subgraph isomorphic to <jats:italic>K<\/jats:italic><jats:sub>1,<jats:italic>r<\/jats:italic><\/jats:sub>. In this paper, we give a minimum degree condition for the <jats:italic>K<\/jats:italic><jats:sub>1,<jats:italic>r<\/jats:italic><\/jats:sub>-free graph with an <jats:italic>S<jats:sub>n<\/jats:sub><\/jats:italic>-factor and the <jats:italic>K<\/jats:italic><jats:sub>1,<jats:italic>r<\/jats:italic><\/jats:sub>-free graph with a <jats:italic>\u2131<\/jats:italic>\u2265<jats:sub>3<\/jats:sub>-factor, respectively. Further, we obtain sufficient conditions for <jats:italic>K<\/jats:italic><jats:sub>1,<jats:italic>r<\/jats:italic><\/jats:sub>-free graphs to be <jats:italic>\u2131<\/jats:italic>\u2265<jats:sub>2<\/jats:sub>-factor, <jats:italic>\u2131<\/jats:italic>\u2265<jats:sub>3<\/jats:sub>-factor or {<jats:italic>P<\/jats:italic><jats:sub>2<\/jats:sub>,\u00a0<jats:italic>P<\/jats:italic><jats:sub>3<\/jats:sub>}-factor covered graphs. In addition, examples show that our results are sharp.<\/jats:p>","DOI":"10.1051\/ro\/2023042","type":"journal-article","created":{"date-parts":[[2023,3,30]],"date-time":"2023-03-30T19:03:44Z","timestamp":1680203024000},"page":"837-846","source":"Crossref","is-referenced-by-count":1,"title":["Remarks on component factors in <i>K<\/i><sub>1,<i>r<\/i><\/sub>-free graphs"],"prefix":"10.1051","volume":"57","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5770-6210","authenticated-orcid":false,"given":"Guowei","family":"Dai","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zan-Bo","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiaoyan","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"250","published-online":{"date-parts":[[2023,4,28]]},"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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