{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,21]],"date-time":"2026-01-21T04:18:30Z","timestamp":1768969110213,"version":"3.49.0"},"reference-count":25,"publisher":"EDP Sciences","issue":"6","license":[{"start":{"date-parts":[[2022,11,29]],"date-time":"2022-11-29T00:00:00Z","timestamp":1669680000000},"content-version":"vor","delay-in-days":28,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["RAIRO-Oper. Res."],"accepted":{"date-parts":[[2022,11,11]]},"published-print":{"date-parts":[[2022,11]]},"abstract":"<jats:p>A path-factor is a spanning subgraph<jats:italic>F<\/jats:italic>of<jats:italic>G<\/jats:italic>such that each component of<jats:italic>F<\/jats:italic>is a path of order at least two. Let<jats:italic>k<\/jats:italic>be an integer with<jats:italic>k<\/jats:italic>\u00a0\u2265\u00a02. A<jats:italic>P<\/jats:italic><jats:sub>\u2265<jats:italic>k<\/jats:italic><\/jats:sub>-factor is a spanning subgraph of<jats:italic>G<\/jats:italic>whose components are paths of order at least<jats:italic>k<\/jats:italic>. A graph<jats:italic>G<\/jats:italic>is called a<jats:italic>P<\/jats:italic><jats:sub>\u2265<jats:italic>k<\/jats:italic><\/jats:sub>-factor covered graph if for any edge<jats:italic>e<\/jats:italic>of<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 covering<jats:italic>e<\/jats:italic>. A graph<jats:italic>G<\/jats:italic>is called a<jats:italic>P<\/jats:italic><jats:sub>\u2265<jats:italic>k<\/jats:italic><\/jats:sub>-factor uniform graph if for any two distinct edges<jats:italic>e<\/jats:italic><jats:sub>1<\/jats:sub>and<jats:italic>e<\/jats:italic><jats:sub>2<\/jats:sub>of<jats:italic>G<\/jats:italic>,<jats:italic>G<\/jats:italic>has a<jats:italic>P<\/jats:italic><jats:sub>\u2265<jats:italic>k<\/jats:italic><\/jats:sub>-factor covering<jats:italic>e<\/jats:italic><jats:sub>1<\/jats:sub>and excluding<jats:italic>e<\/jats:italic><jats:sub>2<\/jats:sub>. In this article, we claim that (1) a 4-edge-connected graph<jats:italic>G<\/jats:italic>is a<jats:italic>P<\/jats:italic><jats:sub>\u22653<\/jats:sub>-factor uniform graph if its sun toughness<jats:italic>s<\/jats:italic>(<jats:italic>G<\/jats:italic>)\u00a0\u2265\u00a01; (2) a 4-connected graph<jats:italic>G<\/jats:italic>is a<jats:italic>P<\/jats:italic><jats:sub>\u22653<\/jats:sub>-factor uniform graph if its sun toughness<jats:italic>s<\/jats:italic>(<jats:italic>G<\/jats:italic>)&gt;4\/5.<\/jats:p>","DOI":"10.1051\/ro\/2022201","type":"journal-article","created":{"date-parts":[[2022,11,14]],"date-time":"2022-11-14T20:02:55Z","timestamp":1668456175000},"page":"4057-4062","source":"Crossref","is-referenced-by-count":9,"title":["Sun toughness and path-factor uniform graphs"],"prefix":"10.1051","volume":"56","author":[{"given":"Hongxia","family":"Liu","sequence":"first","affiliation":[]}],"member":"250","published-online":{"date-parts":[[2022,11,29]]},"reference":[{"key":"R1","doi-asserted-by":"crossref","first-page":"106162","DOI":"10.1016\/j.ipl.2021.106162","volume":"172","author":"Gao","year":"2021","journal-title":"Inf. Process. Lett."},{"key":"R2","first-page":"303","volume":"105","author":"Gao","year":"2017","journal-title":"Util. Math."},{"key":"R3","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1016\/S0095-8956(03)00027-3","volume":"88","author":"Kaneko","year":"2003","journal-title":"J. Combin. Theory Ser. B"},{"key":"R4","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1016\/j.disc.2004.01.016","volume":"283","author":"Kano","year":"2004","journal-title":"Discrete Math."},{"key":"R5","doi-asserted-by":"crossref","first-page":"385","DOI":"10.1016\/j.aml.2009.11.003","volume":"23","author":"Kano","year":"2010","journal-title":"Appl. Math. Lett."},{"key":"R6","doi-asserted-by":"crossref","first-page":"270","DOI":"10.1134\/S0032946020030047","volume":"56","author":"Wang","year":"2020","journal-title":"Probl. Inf. Transm."},{"key":"R7","doi-asserted-by":"crossref","first-page":"969","DOI":"10.1051\/ro\/2021037","volume":"55","author":"Wang","year":"2021","journal-title":"RAIRO-Oper. Res."},{"key":"R8","doi-asserted-by":"crossref","first-page":"2613","DOI":"10.1051\/ro\/2022123","volume":"56","author":"Wang","year":"2022","journal-title":"RAIRO-Oper. Res."},{"key":"R9","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1016\/0095-8956(73)90038-5","volume":"15","author":"Woodall","year":"1973","journal-title":"J. Combin. Theory Ser. B"},{"key":"R10","first-page":"385","volume":"16","author":"Yang","year":"2001","journal-title":"Appl. Math. A J. Chin. Univ. Ser. A"},{"key":"R11","doi-asserted-by":"crossref","first-page":"183","DOI":"10.7151\/dmgt.2075","volume":"39","author":"Yuan","year":"2019","journal-title":"Discuss. Math. Graph Theory"},{"key":"R12","doi-asserted-by":"crossref","first-page":"2067","DOI":"10.1016\/j.disc.2008.04.022","volume":"309","author":"Zhang","year":"2009","journal-title":"Discrete Math."},{"key":"R13","doi-asserted-by":"crossref","unstructured":"Zhou S., Path factors and neighborhoods of independent sets in graphs. Acta Math. Appl. Sin. Engl. Ser. (2022) DOI: 10.1007\/s10255-022-1096-2.","DOI":"10.1007\/s10255-022-1096-2"},{"key":"R14","doi-asserted-by":"crossref","first-page":"657","DOI":"10.1007\/s10255-021-1034-8","volume":"37","author":"Zhou","year":"2021","journal-title":"Acta Math. Appl. Sin. Engl. Ser."},{"key":"R15","doi-asserted-by":"crossref","unstructured":"Zhou S., Remarks on restricted fractional (g,\u00a0f)-factors in graphs. Discrete Appl. Math. (2022) DOI: 10.1016\/j.dam.2022.07.020.","DOI":"10.1016\/j.dam.2022.07.020"},{"key":"R16","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1016\/j.dam.2021.05.022","volume":"323","author":"Zhou","year":"2022","journal-title":"Discrete Appl. Math."},{"key":"R17","doi-asserted-by":"crossref","first-page":"61","DOI":"10.3233\/FI-222130","volume":"187","author":"Zhou","year":"2022","journal-title":"Fundam. Inform."},{"key":"R18","doi-asserted-by":"crossref","first-page":"111715","DOI":"10.1016\/j.disc.2019.111715","volume":"343","author":"Zhou","year":"2020","journal-title":"Discrete Math."},{"key":"R19","doi-asserted-by":"crossref","first-page":"2919","DOI":"10.1051\/ro\/2022143","volume":"56","author":"Zhou","year":"2022","journal-title":"RAIRO-Oper. Res."},{"key":"R20","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1016\/j.dam.2021.04.012","volume":"319","author":"Zhou","year":"2022","journal-title":"Discrete Appl. Math."},{"key":"R21","doi-asserted-by":"crossref","first-page":"511","DOI":"10.1016\/j.dam.2021.03.004","volume":"319","author":"Zhou","year":"2022","journal-title":"Discrete Appl. Math."},{"key":"R22","doi-asserted-by":"crossref","unstructured":"Zhou S., Sun Z. and Bian Q., Isolated toughness and path-factor uniform graphs (II). Indian J. Pure Appl. Math. (2022) DOI: 10.1007\/s13226-022-00286-x.","DOI":"10.1007\/s13226-022-00286-x"},{"key":"R23","first-page":"3","volume":"23","author":"Zhou","year":"2022","journal-title":"Proc. Rom. Acad. Ser. A: Math. Phys. Tech. Sci. Inf. Sci."},{"key":"R24","doi-asserted-by":"crossref","first-page":"795","DOI":"10.1007\/s00010-021-00852-4","volume":"96","author":"Zhou","year":"2022","journal-title":"Aequationes Math."},{"key":"R25","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1017\/S0004972721000952","volume":"106","author":"Zhou","year":"2022","journal-title":"Bull. Aust. Math. Soc."}],"container-title":["RAIRO - Operations Research"],"original-title":[],"link":[{"URL":"https:\/\/www.rairo-ro.org\/10.1051\/ro\/2022201\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,11,30]],"date-time":"2023-11-30T20:25:56Z","timestamp":1701375956000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.rairo-ro.org\/10.1051\/ro\/2022201"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,11]]},"references-count":25,"journal-issue":{"issue":"6"},"alternative-id":["ro220590"],"URL":"https:\/\/doi.org\/10.1051\/ro\/2022201","relation":{},"ISSN":["0399-0559","2804-7303"],"issn-type":[{"value":"0399-0559","type":"print"},{"value":"2804-7303","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,11]]}}}