{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,10]],"date-time":"2026-03-10T03:34:05Z","timestamp":1773113645069,"version":"3.50.1"},"reference-count":31,"publisher":"EDP Sciences","issue":"6","license":[{"start":{"date-parts":[[2025,12,10]],"date-time":"2025-12-10T00:00:00Z","timestamp":1765324800000},"content-version":"vor","delay-in-days":39,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12201408, 12271362"],"award-info":[{"award-number":["12201408, 12271362"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["RAIRO-Oper. Res."],"accepted":{"date-parts":[[2025,10,23]]},"published-print":{"date-parts":[[2025,11]]},"abstract":"<jats:p>\n                    For a graph\n                    <jats:italic>G<\/jats:italic>\n                    and a set\n                    <jats:italic>H<\/jats:italic>\n                    of connected graphs, an\n                    <jats:italic>H<\/jats:italic>\n                    -factor of\n                    <jats:italic>G<\/jats:italic>\n                    is a spanning subgraph of\n                    <jats:italic>G<\/jats:italic>\n                    with each component isomorphic to some member in\n                    <jats:italic>H<\/jats:italic>\n                    . If each component of\n                    <jats:italic>H<\/jats:italic>\n                    is isomorphic to a path, then we call the\n                    <jats:italic>H<\/jats:italic>\n                    -factor a path-factor. For each integer\n                    <jats:italic>k \u2265<\/jats:italic>\n                    2, a graph\n                    <jats:italic>G<\/jats:italic>\n                    is\n                    <jats:italic>\n                      P\n                      <jats:sub>\u2265<\/jats:sub>\n                      <jats:sub>k<\/jats:sub>\n                    <\/jats:italic>\n                    -factor uniform if for any two distinct edges\n                    <jats:italic>e<\/jats:italic>\n                    <jats:sub>1<\/jats:sub>\n                    and\n                    <jats:italic>e<\/jats:italic>\n                    <jats:sub>2<\/jats:sub>\n                    ,\n                    <jats:italic>G<\/jats:italic>\n                    admits a\n                    <jats:italic>\n                      P\n                      <jats:sub>\u2265<\/jats:sub>\n                      <jats:sub>k<\/jats:sub>\n                    <\/jats:italic>\n                    -factor including\n                    <jats:italic>e<\/jats:italic>\n                    <jats:sub>1<\/jats:sub>\n                    and excluding\n                    <jats:italic>e<\/jats:italic>\n                    <jats:sub>2<\/jats:sub>\n                    . In this note, we determine two lower bounds on the degree sum of non-adjacent vertices to ensure that\n                    <jats:italic>G<\/jats:italic>\n                    is\n                    <jats:italic>\n                      P\n                      <jats:sub>\u2265<\/jats:sub>\n                      <jats:sub>k<\/jats:sub>\n                    <\/jats:italic>\n                    -factor uniform for\n                    <jats:italic>k<\/jats:italic>\n                    = 2 and\n                    <jats:italic>k<\/jats:italic>\n                    = 3. Furthermore, we construct some extremal graphs to show that the bounds are best possible. The results improve some known results slightly.\n                  <\/jats:p>","DOI":"10.1051\/ro\/2025148","type":"journal-article","created":{"date-parts":[[2025,10,29]],"date-time":"2025-10-29T08:55:48Z","timestamp":1761728148000},"page":"3675-3681","source":"Crossref","is-referenced-by-count":1,"title":["Characterizing path-factor uniform graphs with respect to the degree sum of non-adjacent vertices"],"prefix":"10.1051","volume":"59","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0171-5978","authenticated-orcid":false,"given":"Ping","family":"Zhang","sequence":"first","affiliation":[{"name":"College of Science, University of Shanghai for Science and Technology","place":["P.R. China"]}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"250","published-online":{"date-parts":[[2025,12,10]]},"reference":[{"key":"R1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1002\/jgt.3190090103","volume":"9","author":"Akiyama","year":"1985","journal-title":"J. Graph Theory"},{"key":"R2","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":"R3","doi-asserted-by":"crossref","first-page":"721","DOI":"10.1051\/ro\/2022033","volume":"56","author":"Dai","year":"2022","journal-title":"RAIRO Oper. Res."},{"key":"R4","doi-asserted-by":"crossref","first-page":"5","DOI":"10.7151\/dmgt.2353","volume":"43","author":"Dai","year":"2023","journal-title":"Discuss. Math. Graph Theory"},{"key":"R5","doi-asserted-by":"crossref","first-page":"1409","DOI":"10.1007\/s13226-023-00446-7","volume":"55","author":"Dai","year":"2024","journal-title":"Indian J. Pure Appl. Math."},{"key":"R6","doi-asserted-by":"crossref","first-page":"1913","DOI":"10.1007\/s00373-020-02184-7","volume":"36","author":"Dai","year":"2020","journal-title":"Graphs Comb."},{"key":"R7","doi-asserted-by":"crossref","first-page":"2907","DOI":"10.1051\/ro\/2021140","volume":"55","author":"Dai","year":"2021","journal-title":"RAIRO Oper. Res."},{"key":"R8","first-page":"1","volume":"25","author":"Egawa","year":"2018","journal-title":"Electron. J. Comb."},{"key":"R9","doi-asserted-by":"crossref","first-page":"106162","DOI":"10.1016\/j.ipl.2021.106162","volume":"172","author":"Gao","year":"2021","journal-title":"Inform. Process. Lett."},{"key":"R10","doi-asserted-by":"crossref","first-page":"809","DOI":"10.1007\/s12190-020-01462-0","volume":"66","author":"Hua","year":"2021","journal-title":"J. Appl. Math. Comput."},{"key":"R11","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1016\/S0095-8956(03)00027-3","volume":"88","author":"Kaneko","year":"2003","journal-title":"J. Comb. Theory Ser. B"},{"key":"R12","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1002\/1097-0118(200104)36:4<175::AID-JGT1005>3.0.CO;2-T","volume":"36","author":"Kaneko","year":"2001","journal-title":"J. Graph Theory"},{"key":"R13","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":"R14","doi-asserted-by":"crossref","first-page":"188","DOI":"10.1002\/jgt.10022","volume":"39","author":"Kawarabayashi","year":"2002","journal-title":"J. Graph Theory"},{"key":"R15","doi-asserted-by":"crossref","first-page":"112588","DOI":"10.1016\/j.disc.2021.112588","volume":"344","author":"Li","year":"2021","journal-title":"Discrete Math."},{"key":"R16","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1016\/j.dam.2024.07.043","volume":"359","author":"Liu","year":"2024","journal-title":"Discrete Appl. Math."},{"key":"R17","doi-asserted-by":"crossref","first-page":"791","DOI":"10.1016\/j.disc.2005.11.059","volume":"307","author":"Plummer","year":"2007","journal-title":"Discrete Math."},{"key":"R18","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1112\/jlms\/s1-22.2.107","volume":"22","author":"Tutte","year":"1947","journal-title":"J. London Math. Soc."},{"key":"R19","doi-asserted-by":"crossref","first-page":"2613","DOI":"10.1051\/ro\/2022123","volume":"56","author":"Wang","year":"2022","journal-title":"RAIRO Oper. Res."},{"key":"R20","doi-asserted-by":"crossref","first-page":"2231","DOI":"10.1051\/ro\/2023111","volume":"57","author":"Wang","year":"2023","journal-title":"RAIRO Oper. Res."},{"key":"R21","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1016\/0095-8956(73)90038-5","volume":"15","author":"Woodall","year":"1973","journal-title":"J. Comb. Theory Ser. B"},{"key":"R22","doi-asserted-by":"crossref","unstructured":"Yu Q. and Liu G., Graph Factors and Matching Extensions. Higher Education Press, Beijing (2009).","DOI":"10.1007\/978-3-540-93952-8"},{"key":"R23","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":"R24","first-page":"87","volume":"97","author":"Zhang","year":"2010","journal-title":"ARS Comb."},{"key":"R25","doi-asserted-by":"crossref","first-page":"4521","DOI":"10.1051\/ro\/2024184","volume":"58","author":"Zhang","year":"2024","journal-title":"RAIRO Oper. Res."},{"key":"R26","doi-asserted-by":"crossref","first-page":"124","DOI":"10.1007\/s40840-025-01902-9","volume":"48","author":"Zhang","year":"2025","journal-title":"Bull. Malays. Math. Sci. Soc."},{"key":"R27","doi-asserted-by":"crossref","first-page":"124","DOI":"10.1016\/j.dam.2025.03.014","volume":"369","author":"Zhou","year":"2025","journal-title":"Discrete Appl. Math."},{"key":"R28","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":"R29","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":"R30","doi-asserted-by":"crossref","first-page":"489","DOI":"10.1007\/s00010-023-00944-3","volume":"97","author":"Zhou","year":"2023","journal-title":"Aequationes Math."},{"key":"R31","doi-asserted-by":"crossref","first-page":"113940","DOI":"10.1016\/j.disc.2024.113940","volume":"347","author":"Zhou","year":"2024","journal-title":"Discrete Math."}],"container-title":["RAIRO - Operations Research"],"original-title":[],"link":[{"URL":"https:\/\/www.rairo-ro.org\/10.1051\/ro\/2025148\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,12,10]],"date-time":"2025-12-10T09:00:09Z","timestamp":1765357209000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.rairo-ro.org\/10.1051\/ro\/2025148"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,11]]},"references-count":31,"journal-issue":{"issue":"6"},"alternative-id":["ro240771"],"URL":"https:\/\/doi.org\/10.1051\/ro\/2025148","relation":{},"ISSN":["0399-0559","2804-7303"],"issn-type":[{"value":"0399-0559","type":"print"},{"value":"2804-7303","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,11]]}}}