{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,28]],"date-time":"2025-09-28T12:48:35Z","timestamp":1759063715122},"reference-count":17,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2006,10,3]],"date-time":"2006-10-03T00:00:00Z","timestamp":1159833600000},"content-version":"vor","delay-in-days":7886,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Graph Theory"],"published-print":{"date-parts":[[1985,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Let <jats:italic>a<\/jats:italic> and <jats:italic>b<\/jats:italic> be integers such that 0 \u2a7d <jats:italic>a<\/jats:italic> \u2a7d <jats:italic>b<\/jats:italic>. Then a graph <jats:italic>G<\/jats:italic> is called an [<jats:italic>a<\/jats:italic>, <jats:italic>b<\/jats:italic>]\u2010graph if a \u2a7d <jats:italic>d<\/jats:italic><jats:sub>G<\/jats:sub>(x) \u03f5 <jats:italic>b<\/jats:italic> for every x \u03f5 <jats:italic>V<\/jats:italic>(G), and an [a, b]\u2010factor of a graph is defined to be its spanning subgraph <jats:italic>F<\/jats:italic> such that <jats:italic>a<\/jats:italic> \u2a7d <jats:italic>d<\/jats:italic><jats:sub>F<\/jats:sub>(x) \u2a7d <jats:italic>b<\/jats:italic> for every vertex x, where <jats:italic>d<\/jats:italic><jats:sub>G<\/jats:sub>(x) and <jats:italic>d<\/jats:italic><jats:sub>F<\/jats:sub>(x) denote the degrees of x in <jats:italic>G<\/jats:italic> and <jats:italic>F<\/jats:italic>, respectively. If the edges of a graph can be decomposed into [<jats:italic>a.b<\/jats:italic>]\u2010factors then we say that the graph is [2<jats:italic>a<\/jats:italic>, <jats:italic>2a<\/jats:italic>]\u2010factorable. We prove the following two theorems: (i) a graph <jats:italic>G<\/jats:italic> is [2<jats:italic>a<\/jats:italic>, <jats:italic>2b<\/jats:italic>)\u2010factorable if and only if <jats:italic>G<\/jats:italic> is a [2<jats:italic>am,2bm<\/jats:italic>]\u2010graph for some integer <jats:italic>m<\/jats:italic>, and (ii) every [8<jats:italic>m<\/jats:italic> + 2k, 10<jats:italic>m<\/jats:italic> + 2<jats:italic>k<\/jats:italic>]\u2010graph is [1,2]\u2010factorable.<\/jats:p>","DOI":"10.1002\/jgt.3190090111","type":"journal-article","created":{"date-parts":[[2007,5,29]],"date-time":"2007-05-29T07:47:11Z","timestamp":1180424831000},"page":"129-146","source":"Crossref","is-referenced-by-count":29,"title":["[<i>a<\/i>,<i>b<\/i>]\u2010factorization of a graph"],"prefix":"10.1002","volume":"9","author":[{"given":"Mikio","family":"Kano","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2006,10,3]]},"reference":[{"key":"e_1_2_1_2_2","unstructured":"J.Akiyama Factorization and linear arboricity of graphs Doctoral thesis Science University of Tokyo February (1982)."},{"key":"e_1_2_1_3_2","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.3190090103"},{"key":"e_1_2_1_4_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF01214296"},{"key":"e_1_2_1_5_2","volume-title":"Graphs & Digraphs","author":"Behzad M.","year":"1979"},{"key":"e_1_2_1_6_2","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.3190090107"},{"key":"e_1_2_1_7_2","volume-title":"Proceedings of the First Colorado Symposium of Graph Theory","author":"Era H."},{"key":"e_1_2_1_8_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF02022560"},{"key":"e_1_2_1_9_2","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365X(83)90077-8"},{"key":"e_1_2_1_10_2","series-title":"Lecture Notes in Mathematics 1073","volume-title":"Graph factors with given properties. Graph Theory Singapore 1983","author":"Kano M.","year":"1984"},{"key":"e_1_2_1_11_2","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365X(78)90006-7"},{"key":"e_1_2_1_12_2","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365X(75)90085-0"},{"key":"e_1_2_1_12_3","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365X(76)90061-3"},{"key":"e_1_2_1_13_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0021-9800(70)80033-3"},{"key":"e_1_2_1_14_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF02392606"},{"key":"e_1_2_1_15_2","first-page":"310","article-title":"Connectivity of regular graphs and the existence of 1\u2010factors","volume":"22","author":"Plesnik J.","year":"1972","journal-title":"Matematick\u1ef3 casopis"},{"key":"e_1_2_1_16_2","unstructured":"A.SaitoandM.Kano Some results on factors. Technical Report 82\u201310 Department of Information Science Tokyo University (1982)."},{"key":"e_1_2_1_17_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF02579180"}],"container-title":["Journal of Graph Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fjgt.3190090111","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/jgt.3190090111","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,21]],"date-time":"2023-10-21T05:44:54Z","timestamp":1697867094000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/jgt.3190090111"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1985,3]]},"references-count":17,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1985,3]]}},"alternative-id":["10.1002\/jgt.3190090111"],"URL":"https:\/\/doi.org\/10.1002\/jgt.3190090111","archive":["Portico"],"relation":{},"ISSN":["0364-9024","1097-0118"],"issn-type":[{"value":"0364-9024","type":"print"},{"value":"1097-0118","type":"electronic"}],"subject":[],"published":{"date-parts":[[1985,3]]}}}