{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,21]],"date-time":"2026-01-21T13:24:05Z","timestamp":1769001845583,"version":"3.49.0"},"reference-count":23,"publisher":"EDP Sciences","issue":"6","license":[{"start":{"date-parts":[[2020,9,16]],"date-time":"2020-09-16T00:00:00Z","timestamp":1600214400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.edpsciences.org\/en\/authors\/copyright-and-licensing"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["RAIRO-Oper. Res."],"accepted":{"date-parts":[[2019,11,20]]},"published-print":{"date-parts":[[2020,11]]},"abstract":"<jats:p>A spanning subgraph of a graph is defined as a path factor of the graph if its component are paths. A <jats:italic>P<\/jats:italic><jats:sub>\u2265<jats:italic>n<\/jats:italic><\/jats:sub>-factor means a path factor with each component having at least <jats:italic>n<\/jats:italic> vertices. A graph <jats:italic>G<\/jats:italic> is defined as a (<jats:italic>P<\/jats:italic><jats:sub>\u2265<jats:italic>n<\/jats:italic><\/jats:sub>, <jats:italic>m<\/jats:italic>)-factor deleted graph if <jats:italic>G<\/jats:italic>\u2013<jats:italic>E<\/jats:italic>\u2032 has a <jats:italic>P<\/jats:italic><jats:sub>\u2265<jats:italic>n<\/jats:italic><\/jats:sub>-factor for every <jats:italic>E<\/jats:italic>\u2032\u00a0\u2286\u00a0<jats:italic>E<\/jats:italic>(<jats:italic>G<\/jats:italic>) with |<jats:italic>E<\/jats:italic>\u2032|\u00a0=\u00a0<jats:italic>m<\/jats:italic>. A graph <jats:italic>G<\/jats:italic> is defined as a (<jats:italic>P<\/jats:italic><jats:sub>\u2265<jats:italic>n<\/jats:italic><\/jats:sub>, <jats:italic>k<\/jats:italic>)-factor critical graph if after deleting any <jats:italic>k<\/jats:italic> vertices of <jats:italic>G<\/jats:italic> the remaining graph of <jats:italic>G<\/jats:italic> admits a <jats:italic>P<\/jats:italic><jats:sub>\u2265<jats:italic>n<\/jats:italic><\/jats:sub>-factor. In this paper, we demonstrate that (i) a graph <jats:italic>G<\/jats:italic> is (<jats:italic>P<\/jats:italic><jats:sub>\u22653<\/jats:sub>, <jats:italic>m<\/jats:italic>)-factor deleted if <jats:italic>\u03ba<\/jats:italic>(<jats:italic>G<\/jats:italic>)\u00a0\u2265\u00a02<jats:italic>m<\/jats:italic>\u00a0+\u00a01 and <jats:italic>bind<\/jats:italic>(<jats:italic>G<\/jats:italic>)\u00a0\u2265\u00a0 2\/3 - $ \\frac{3}{2}-\\frac{1}{4m+4}$; (ii) a graph <jats:italic>G<\/jats:italic> is (<jats:italic>P<\/jats:italic><jats:sub>\u22653<\/jats:sub>, <jats:italic>k<\/jats:italic>)-factor critical if <jats:italic>\u03ba<\/jats:italic>(<jats:italic>G<\/jats:italic>)\u00a0\u2265\u00a0<jats:italic>k<\/jats:italic>\u00a0+\u00a02 and <jats:italic>bind<\/jats:italic>(<jats:italic>G<\/jats:italic>)\u00a0\u2265\u00a0$ \\frac{5+k}{4}$.<\/jats:p>","DOI":"10.1051\/ro\/2019111","type":"journal-article","created":{"date-parts":[[2019,11,22]],"date-time":"2019-11-22T19:58:03Z","timestamp":1574452683000},"page":"1827-1834","source":"Crossref","is-referenced-by-count":32,"title":["Remarks on path factors in graphs"],"prefix":"10.1051","volume":"54","author":[{"given":"Sizhong","family":"Zhou","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"250","published-online":{"date-parts":[[2020,9,16]]},"reference":[{"key":"R1","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1016\/S0012-365X(01)00214-X","volume":"243","author":"Ando","year":"2002","journal-title":"Discrete Math."},{"key":"R2","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1016\/S0012-365X(99)00104-1","volume":"205","author":"Enomoto","year":"1999","journal-title":"Discrete Math."},{"key":"R3","doi-asserted-by":"crossref","first-page":"55","DOI":"10.4064\/cm6713-8-2016","volume":"147","author":"Gao","year":"2017","journal-title":"Colloq. 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