{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,1]],"date-time":"2026-04-01T14:44:37Z","timestamp":1775054677658,"version":"3.50.1"},"reference-count":27,"publisher":"EDP Sciences","issue":"4","license":[{"start":{"date-parts":[[2022,8,18]],"date-time":"2022-08-18T00:00:00Z","timestamp":1660780800000},"content-version":"vor","delay-in-days":48,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["RAIRO-Oper. Res."],"accepted":{"date-parts":[[2022,7,18]]},"published-print":{"date-parts":[[2022,7]]},"abstract":"<jats:p>A graph<jats:italic>G<\/jats:italic>is a fractional (<jats:italic>a<\/jats:italic>,\u00a0<jats:italic>b<\/jats:italic>,\u00a0<jats:italic>k<\/jats:italic>)-critical covered graph if<jats:italic>G<\/jats:italic>\u00a0\u2212\u00a0<jats:italic>U<\/jats:italic>is a fractional [<jats:italic>a<\/jats:italic>,\u00a0<jats:italic>b<\/jats:italic>]-covered graph for every<jats:italic>U<\/jats:italic>\u00a0\u2286\u00a0<jats:italic>V<\/jats:italic>(<jats:italic>G<\/jats:italic>) with |<jats:italic>U<\/jats:italic>|\u00a0=\u00a0<jats:italic>k<\/jats:italic>, which is first defined by (Zhou, Xu and Sun,<jats:italic>Inf. Process. Lett.<\/jats:italic><jats:bold>152<\/jats:bold>(2019) 105838). Furthermore, they derived a degree condition for a graph to be a fractional (<jats:italic>a<\/jats:italic>,\u00a0<jats:italic>b<\/jats:italic>,\u00a0<jats:italic>k<\/jats:italic>)-critical covered graph. In this paper, we gain an independence number and connectivity condition for a graph to be a fractional (<jats:italic>a<\/jats:italic>,\u00a0<jats:italic>b<\/jats:italic>,\u00a0<jats:italic>k<\/jats:italic>)-critical covered graph and verify that<jats:italic>G<\/jats:italic>is a fractional (<jats:italic>a<\/jats:italic>,\u00a0<jats:italic>b<\/jats:italic>,\u00a0<jats:italic>k<\/jats:italic>)-critical covered graph if<\/jats:p><jats:p><jats:italic>k<\/jats:italic>(<jats:italic>G<\/jats:italic>) \u2265 max {2<jats:italic>b<\/jats:italic>(<jats:italic>a<\/jats:italic>+1)(<jats:italic>b<\/jats:italic>+1)+4<jats:italic>bk<\/jats:italic>+5\/4<jats:italic>b<\/jats:italic>,(<jats:italic>a<\/jats:italic>+1)<jats:sup>2<\/jats:sup><jats:italic>\ud835\udefc<\/jats:italic>(<jats:italic>G<\/jats:italic>)+4<jats:italic>bk<\/jats:italic>+5\/4<jats:italic>b<\/jats:italic>}.<\/jats:p>","DOI":"10.1051\/ro\/2022119","type":"journal-article","created":{"date-parts":[[2022,7,19]],"date-time":"2022-07-19T18:53:45Z","timestamp":1658256825000},"page":"2535-2542","source":"Crossref","is-referenced-by-count":16,"title":["Independence number and connectivity for fractional (<i>a<\/i>,\u00a0<i>b<\/i>,\u00a0<i>k<\/i>)-critical covered graphs"],"prefix":"10.1051","volume":"56","author":[{"given":"Sizhong","family":"Zhou","sequence":"first","affiliation":[]},{"given":"Jiancheng","family":"Wu","sequence":"additional","affiliation":[]},{"given":"Hongxia","family":"Liu","sequence":"additional","affiliation":[]}],"member":"250","published-online":{"date-parts":[[2022,8,18]]},"reference":[{"key":"R1","doi-asserted-by":"crossref","first-page":"1249","DOI":"10.1016\/j.ejc.2006.05.003","volume":"27","author":"Assiyatun","year":"2006","journal-title":"Eur. J. Comb."},{"key":"R2","doi-asserted-by":"crossref","first-page":"108","DOI":"10.1016\/j.dam.2013.09.005","volume":"162","author":"Bekkai","year":"2014","journal-title":"Discrete Appl. Math."},{"key":"R3","doi-asserted-by":"crossref","first-page":"281","DOI":"10.1016\/j.dam.2010.12.003","volume":"159","author":"Belcastro","year":"2011","journal-title":"Discrete Appl. Math."},{"key":"R4","doi-asserted-by":"crossref","first-page":"757","DOI":"10.2298\/FIL1504757B","volume":"29","author":"Bian","year":"2015","journal-title":"Filomat"},{"key":"R5","doi-asserted-by":"crossref","first-page":"265","DOI":"10.1016\/j.dam.2020.01.014","volume":"283","author":"Egawa","year":"2020","journal-title":"Discrete Appl. Math."},{"key":"R6","doi-asserted-by":"crossref","first-page":"1227","DOI":"10.1007\/s10114-019-8169-z","volume":"35","author":"Gao","year":"2019","journal-title":"Acta Math. Sin. Engl. Ser."},{"key":"R7","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1007\/s10114-020-9156-0","volume":"36","author":"Gao","year":"2020","journal-title":"Acta Math. Sin. Engl. Ser."},{"key":"R8","doi-asserted-by":"crossref","first-page":"85","DOI":"10.3792\/pjaa.86.85","volume":"86","author":"Kotani","year":"2010","journal-title":"Proc. Japan Acad. Ser. A"},{"key":"R9","first-page":"65","volume":"6","author":"Li","year":"2002","journal-title":"OR Trans. (China)"},{"key":"R10","doi-asserted-by":"crossref","first-page":"2293","DOI":"10.1016\/j.dam.2007.10.016","volume":"156","author":"Liu","year":"2008","journal-title":"Discrete Appl. Math."},{"key":"R11","doi-asserted-by":"crossref","first-page":"872","DOI":"10.3934\/math.2020059","volume":"5","author":"Lv","year":"2020","journal-title":"AIMS Math."},{"key":"R12","doi-asserted-by":"crossref","first-page":"797","DOI":"10.7151\/dmgt.1964","volume":"37","author":"Tsuchiya","year":"2017","journal-title":"Discuss. Math. Graph Theory"},{"key":"R13","doi-asserted-by":"crossref","first-page":"270","DOI":"10.1134\/S0032946020030047","volume":"56","author":"Wang","year":"2020","journal-title":"Probl. Inf. Transm."},{"key":"R14","doi-asserted-by":"crossref","first-page":"969","DOI":"10.1051\/ro\/2021037","volume":"55","author":"Wang","year":"2021","journal-title":"RAIRO-Oper. Res."},{"key":"R15","doi-asserted-by":"crossref","first-page":"775","DOI":"10.1007\/s10255-018-0786-2","volume":"34","author":"Yuan","year":"2018","journal-title":"Acta Math. Appl. Sin. Engl. Ser."},{"key":"R16","doi-asserted-by":"crossref","unstructured":"Zhou S., Some results on path-factor critical avoidable graphs. Discuss. Math. Graph Theory (2020) DOI: 10.7151\/dmgt.2364.","DOI":"10.7151\/dmgt.2364"},{"key":"R17","doi-asserted-by":"crossref","unstructured":"Zhou S., A neighborhood union condition for fractional (a,\u00a0b,\u00a0k)-critical covered graphs. Discrete Appl. Math. (2021) DOI: 10.1016\/j.dam.2021.05.022.","DOI":"10.1016\/j.dam.2021.05.022"},{"key":"R18","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":"R19","doi-asserted-by":"crossref","unstructured":"Zhou S., Path factors and neighborhoods of independent sets in graphs. Acta Math. Appl. Sin. Engl. Ser. DOI: 10.1007\/s10255-022-1096-2.","DOI":"10.1007\/s10255-022-1096-2"},{"key":"R20","doi-asserted-by":"crossref","first-page":"417","DOI":"10.1007\/s10255-022-1086-4","volume":"38","author":"Zhou","year":"2022","journal-title":"Acta Math. Appl. Sin. Engl. Ser."},{"key":"R21","doi-asserted-by":"crossref","first-page":"105838","DOI":"10.1016\/j.ipl.2019.105838","volume":"152","author":"Zhou","year":"2019","journal-title":"Inf. Process. Lett."},{"key":"R22","doi-asserted-by":"crossref","unstructured":"Zhou S., Wu J. and Xu Y., Toughness, isolated toughness and path factors in graphs. Bull. Aust. Math. Soc. (2021) DOI: 10.1017\/S0004972721000952.","DOI":"10.1017\/S0004972721000952"},{"key":"R23","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":"R24","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":"R25","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":"R26","doi-asserted-by":"crossref","first-page":"178","DOI":"10.1007\/s10255-022-1053-0","volume":"38","author":"Zhou","year":"2022","journal-title":"Acta Math. Appl. Sin. Engl. Ser."},{"key":"R27","doi-asserted-by":"crossref","first-page":"795","DOI":"10.1007\/s00010-021-00852-4","volume":"96","author":"Zhou","year":"2022","journal-title":"Aequationes Math."}],"container-title":["RAIRO - Operations Research"],"original-title":[],"link":[{"URL":"https:\/\/www.rairo-ro.org\/10.1051\/ro\/2022119\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,11,24]],"date-time":"2023-11-24T19:21:30Z","timestamp":1700853690000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.rairo-ro.org\/10.1051\/ro\/2022119"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,7]]},"references-count":27,"journal-issue":{"issue":"4"},"alternative-id":["ro210198"],"URL":"https:\/\/doi.org\/10.1051\/ro\/2022119","relation":{},"ISSN":["0399-0559","2804-7303"],"issn-type":[{"value":"0399-0559","type":"print"},{"value":"2804-7303","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,7]]}}}