{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,17]],"date-time":"2025-11-17T14:29:09Z","timestamp":1763389749277},"reference-count":35,"publisher":"EDP Sciences","issue":"5","license":[{"start":{"date-parts":[[2023,9,29]],"date-time":"2023-09-29T00:00:00Z","timestamp":1695945600000},"content-version":"vor","delay-in-days":28,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"Natural Science Foundation of Shandong Province, China","award":["ZR2023MA078"],"award-info":[{"award-number":["ZR2023MA078"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["RAIRO-Oper. Res."],"accepted":{"date-parts":[[2023,9,9]]},"published-print":{"date-parts":[[2023,9]]},"abstract":"<jats:p>A graph <jats:italic>G<\/jats:italic> has a strong parity factor <jats:italic>F<\/jats:italic> if for every subset <jats:italic>X<\/jats:italic> \u2286 <jats:italic>V<\/jats:italic> (<jats:italic>G<\/jats:italic>) with |<jats:italic>X<\/jats:italic>| even, <jats:italic>G<\/jats:italic> contains a spanning subgraph <jats:italic>F<\/jats:italic> such that <jats:italic>\u03b4<\/jats:italic>(<jats:italic>F<\/jats:italic>) \u2265 1, <jats:italic>d<\/jats:italic><jats:sub><jats:italic>F<\/jats:italic><\/jats:sub> (<jats:italic>u<\/jats:italic>)\u00a0\u2261\u00a01 (mod 2) for any <jats:italic>u<\/jats:italic>\u00a0\u2208\u00a0<jats:italic>X<\/jats:italic>, and <jats:italic>d<\/jats:italic><jats:sub><jats:italic>F<\/jats:italic><\/jats:sub>(<jats:italic>v<\/jats:italic>)\u00a0\u2261\u00a00 (mod 2) for any <jats:italic>v<\/jats:italic>\u00a0\u2208\u00a0<jats:italic>V<\/jats:italic>(<jats:italic>G<\/jats:italic>)\u00a0\\\u00a0<jats:italic>X<\/jats:italic>. In this article, we first provide a size condition for a graph having a strong parity factor. Then we put forward a toughness condition to guarantee that a graph has a strong parity factor.<\/jats:p>","DOI":"10.1051\/ro\/2023147","type":"journal-article","created":{"date-parts":[[2023,9,12]],"date-time":"2023-09-12T19:03:10Z","timestamp":1694545390000},"page":"2465-2471","source":"Crossref","is-referenced-by-count":2,"title":["Sufficient conditions for graphs to have strong parity factors"],"prefix":"10.1051","volume":"57","author":[{"given":"Sizhong","family":"Zhou","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yuli","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"250","published-online":{"date-parts":[[2023,9,29]]},"reference":[{"key":"R1","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1007\/BF02582935","volume":"1","author":"Amahashi","year":"1985","journal-title":"Graphs Comb."},{"key":"R2","doi-asserted-by":"crossref","first-page":"1391","DOI":"10.1007\/s00373-020-02225-1","volume":"36","author":"Bujt\u00e1s","year":"2020","journal-title":"Graphs Comb."},{"key":"R3","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1016\/0012-365X(73)90138-6","volume":"5","author":"Chv\u00e1tal","year":"1973","journal-title":"Discrete Math."},{"key":"R4","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1002\/jgt.3190120305","volume":"12","author":"Cui","year":"1988","journal-title":"J. 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