{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,10,26]],"date-time":"2023-10-26T09:42:21Z","timestamp":1698313341405},"reference-count":6,"publisher":"Wiley","issue":"3","license":[{"start":{"date-parts":[[2006,10,11]],"date-time":"2006-10-11T00:00:00Z","timestamp":1160524800000},"content-version":"vor","delay-in-days":4911,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Networks"],"published-print":{"date-parts":[[1993,5]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Let <jats:italic>G<\/jats:italic> = <jats:italic>(A, B; E)<\/jats:italic> be a bipartite graph. Let <jats:italic>e<\/jats:italic><jats:sub>1<\/jats:sub>, <jats:italic>e<\/jats:italic><jats:sub>2<\/jats:sub> be nonnegative integers, and <jats:italic>f<\/jats:italic><jats:sub>1<\/jats:sub>, <jats:italic>f<\/jats:italic><jats:sub>2<\/jats:sub> nonnegative integer\u2010valued functions on <jats:italic>V(G)<\/jats:italic> such that <jats:italic>e<\/jats:italic><jats:sub><jats:italic>i<\/jats:italic><\/jats:sub> \u2266 |<jats:italic>E<\/jats:italic>| \u2266 <jats:italic>e<\/jats:italic><jats:sub>1<\/jats:sub> + <jats:italic>e<\/jats:italic><jats:sub>2<\/jats:sub> and <jats:italic>f<jats:sub>i<\/jats:sub>(v)<\/jats:italic> \u2266 <jats:italic>d(v)<\/jats:italic> \u2266 <jats:italic>f<\/jats:italic><jats:sub>1<\/jats:sub><jats:italic>(v)<\/jats:italic> + <jats:italic>f<\/jats:italic><jats:sub>2<\/jats:sub><jats:italic>(v)<\/jats:italic> for all <jats:italic>v<\/jats:italic> \u03f5 <jats:italic>V(G)<\/jats:italic> (<jats:italic>i<\/jats:italic> = 1, 2). Necessary and sufficient conditions are obtained for <jats:italic>G<\/jats:italic> to admit a decomposition in spanning subgraphs <jats:italic>G<\/jats:italic><jats:sub>1<\/jats:sub> = (<jats:italic>A, B; E<\/jats:italic><jats:sub>1<\/jats:sub>) and <jats:italic>G<\/jats:italic><jats:sub>2<\/jats:sub> = (<jats:italic>A, B; E<\/jats:italic><jats:sub>2<\/jats:sub>) such that |<jats:italic>E<\/jats:italic><jats:sub><jats:italic>i<\/jats:italic><\/jats:sub>| \u2266 <jats:italic>e<\/jats:italic><jats:sub><jats:italic>i<\/jats:italic><\/jats:sub> and <jats:italic>d<jats:sub>Gi<\/jats:sub>(v)<\/jats:italic> \u2266 <jats:italic>f<jats:sub>i<\/jats:sub>(v)<\/jats:italic> for all <jats:italic>v<\/jats:italic> \u03f5 <jats:italic>V(G)<\/jats:italic> (<jats:italic>i<\/jats:italic> = 1, 2). The result generalizes a known characterization of bipartite graphs with a <jats:italic>k<\/jats:italic>\u2010factor. Its proof uses flow theory and is a refinement of the proof of an analogous result due to Folkman and Fulkerson. By applying corresponding flow algorithms, the described decomposition can be found in polynomial time if it exists. As an application, an assignment problem is solved. \u00a9 <jats:italic>1993 by John Wiley &amp; Sons, Inc.<\/jats:italic><\/jats:p>","DOI":"10.1002\/net.3230230303","type":"journal-article","created":{"date-parts":[[2007,5,12]],"date-time":"2007-05-12T14:18:03Z","timestamp":1178979483000},"page":"159-164","source":"Crossref","is-referenced-by-count":0,"title":["Decomposition of bipartite graphs under degree constraints"],"prefix":"10.1002","volume":"23","author":[{"given":"H. J.","family":"Broersma","sequence":"first","affiliation":[]},{"given":"R. J.","family":"Faudree","sequence":"additional","affiliation":[]},{"given":"J. Den","family":"Van Heuvel","sequence":"additional","affiliation":[]},{"given":"H. J.","family":"Veldman","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2006,10,11]]},"reference":[{"key":"e_1_2_1_2_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-349-03521-2"},{"key":"e_1_2_1_3_2","first-page":"561","volume-title":"Combinatorial Mathematics and Its Applications","author":"Folkman J.","year":"1969"},{"key":"e_1_2_1_4_2","doi-asserted-by":"publisher","DOI":"10.1515\/9781400875184"},{"key":"e_1_2_1_5_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF01787577"},{"key":"e_1_2_1_6_2","article-title":"Matching theory","volume":"29","author":"Lov\u00e1sz L.","year":"1986","journal-title":"Ann. of Discrete Math."},{"key":"e_1_2_1_7_2","unstructured":"E. A.van Doorn personal communication."}],"container-title":["Networks"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fnet.3230230303","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/net.3230230303","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,25]],"date-time":"2023-10-25T13:14:31Z","timestamp":1698239671000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/net.3230230303"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1993,5]]},"references-count":6,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1993,5]]}},"alternative-id":["10.1002\/net.3230230303"],"URL":"https:\/\/doi.org\/10.1002\/net.3230230303","archive":["Portico"],"relation":{},"ISSN":["0028-3045","1097-0037"],"issn-type":[{"value":"0028-3045","type":"print"},{"value":"1097-0037","type":"electronic"}],"subject":[],"published":{"date-parts":[[1993,5]]}}}