{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,10,26]],"date-time":"2023-10-26T05:40:35Z","timestamp":1698298835995},"reference-count":6,"publisher":"Wiley","issue":"3","license":[{"start":{"date-parts":[[2006,10,5]],"date-time":"2006-10-05T00:00:00Z","timestamp":1160006400000},"content-version":"vor","delay-in-days":4844,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Graph Theory"],"published-print":{"date-parts":[[1993,7]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>A <jats:italic>rooted graph<\/jats:italic> is a pair (<jats:italic>G,x<\/jats:italic>), where <jats:italic>G<\/jats:italic> is a simple undirected graph and <jats:italic>x<\/jats:italic> \u2208 <jats:italic>V<\/jats:italic>(<jats:italic>G<\/jats:italic>). If <jats:italic>G<\/jats:italic> is rooted at <jats:italic>x<\/jats:italic>, its k<jats:italic>th rotation number h<jats:sub>k<\/jats:sub><\/jats:italic> (<jats:italic>G,x<\/jats:italic>) is the minimum number of edges in a graph <jats:italic>F<\/jats:italic> of order |<jats:italic>G<\/jats:italic>| + <jats:italic>k<\/jats:italic> such that for every <jats:italic>v<\/jats:italic> \u2208 <jats:italic>V<\/jats:italic>(<jats:italic>F<\/jats:italic>) we can find a copy of <jats:italic>G<\/jats:italic> in <jats:italic>F<\/jats:italic> with the root vertex <jats:italic>x<\/jats:italic> at <jats:italic>v.<\/jats:italic> When <jats:italic>k<\/jats:italic> = 0, this definition reduces to that of the <jats:italic>rotation number h<\/jats:italic>(<jats:italic>G,x<\/jats:italic>), which was introduced in [\u201cOn Rotation Numbers for Complete Bipartite Graphs,\u201d University of Victoria, Department of Mathematics Report No. DM\u2010186\u2010IR (1979)] by E.J. Cockayne and P.J. Lorimer and subsequently calculated for complete multipartite graphs. In this paper, we estimate the <jats:italic>k<\/jats:italic>th rotation number for complete bipartite graphs <jats:italic>G<\/jats:italic> with root <jats:italic>x<\/jats:italic> in the larger vertex class, thereby generalizing results of B. Bollob\u00e1s and E.J. Cockayne [\u201cMore Rotation Numbers for Complete Bipartite Graphs,\u201d <jats:italic>Journal of Graph Theory<\/jats:italic>, Vol. 6 (1982), pp. 403\u2013411], J. Haviland [\u201cCliques and Independent Sets,\u201d Ph. D. thesis, University of Cambridge (1989)], and J. Haviland and A. Thomason [\u201cRotation Numbers for Complete Bipartite Graphs,\u201d <jats:italic>Journal of Graph Theory<\/jats:italic>, Vol. 16 (1992), pp. 61\u201371]. \u00a9 1993 John Wiley &amp; Sons, Inc.<\/jats:p>","DOI":"10.1002\/jgt.3190170304","type":"journal-article","created":{"date-parts":[[2007,6,7]],"date-time":"2007-06-07T18:22:14Z","timestamp":1181240534000},"page":"291-301","source":"Crossref","is-referenced-by-count":0,"title":["Generalized Rotation numbers"],"prefix":"10.1002","volume":"17","author":[{"given":"Robin J.","family":"Chapman","sequence":"first","affiliation":[]},{"given":"Julie","family":"Haviland","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2006,10,5]]},"reference":[{"key":"e_1_2_1_2_2","volume-title":"Extremal Graph Theory","author":"Bollob\u00e1s B.","year":"1978"},{"key":"e_1_2_1_3_2","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.3190060404"},{"key":"e_1_2_1_4_2","unstructured":"E. J.CockayneandP. J.Lorimer On rotation numbers for complete bipartite graphs. University of Victoria Department of Mathematics Report DM\u2010186\u2010IR (1979)."},{"key":"e_1_2_1_5_2","unstructured":"J.Haviland Cliques and independent sets. Ph. D. thesis University of Cambridge (1989)."},{"key":"e_1_2_1_6_2","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.3190160107"},{"key":"e_1_2_1_7_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF01788140"}],"container-title":["Journal of Graph Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fjgt.3190170304","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/jgt.3190170304","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,25]],"date-time":"2023-10-25T12:08:42Z","timestamp":1698235722000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/jgt.3190170304"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1993,7]]},"references-count":6,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1993,7]]}},"alternative-id":["10.1002\/jgt.3190170304"],"URL":"https:\/\/doi.org\/10.1002\/jgt.3190170304","archive":["Portico"],"relation":{},"ISSN":["0364-9024","1097-0118"],"issn-type":[{"value":"0364-9024","type":"print"},{"value":"1097-0118","type":"electronic"}],"subject":[],"published":{"date-parts":[[1993,7]]}}}