{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,17]],"date-time":"2026-04-17T07:26:52Z","timestamp":1776410812837,"version":"3.51.2"},"reference-count":14,"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":9712,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Graph Theory"],"published-print":{"date-parts":[[1980,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Let <jats:italic>G(n, d)<\/jats:italic> denote a connected regular bipartite graph on 2<jats:italic>n<\/jats:italic> vertices and of degree <jats:italic>d.<\/jats:italic> It is proved that any Cartesian product <jats:italic>G(n, d)<\/jats:italic> \u00d7 <jats:italic>G<\/jats:italic><jats:sub>1<\/jats:sub>(<jats:italic>n<\/jats:italic><jats:sub>1<\/jats:sub>, <jats:italic>d<\/jats:italic><jats:sub>1<\/jats:sub>) \u00d7 <jats:italic>G<\/jats:italic><jats:sub>2<\/jats:sub>(<jats:italic>n<\/jats:italic><jats:sub>2<\/jats:sub>, <jats:italic>d<\/jats:italic><jats:sub>2<\/jats:sub>) \u00d7  \u20db \u00d7 <jats:italic>G<\/jats:italic><jats:sub><jats:italic>m<\/jats:italic><\/jats:sub>(<jats:italic>n<\/jats:italic><jats:sub><jats:italic>m<\/jats:italic><\/jats:sub>, <jats:italic>d<\/jats:italic><jats:sub><jats:italic>m<\/jats:italic><\/jats:sub>), such that max {<jats:italic>d<\/jats:italic><jats:sub>1<\/jats:sub>, <jats:italic>d<\/jats:italic><jats:sub>2<\/jats:sub>,\u2026, <jats:italic>d<\/jats:italic><jats:sub><jats:italic>m<\/jats:italic><\/jats:sub>} \u2264 <jats:italic>d<\/jats:italic> \u2264 <jats:italic>d<\/jats:italic><jats:sub>1<\/jats:sub> + <jats:italic>d<\/jats:italic><jats:sub>2<\/jats:sub> +  \u20db + <jats:italic>d<\/jats:italic><jats:sub><jats:italic>m<\/jats:italic><\/jats:sub>, has a quadrilateral embedding, thereby establishing its genus, and thereby generalizing a result of White. It is also proved that if <jats:italic>G<\/jats:italic> is any connected bipartite graph of maximum degree <jats:italic>D<\/jats:italic>, if <jats:italic>Q<\/jats:italic><jats:sub><jats:italic>m<\/jats:italic><\/jats:sub> is the <jats:italic>m<\/jats:italic>\u2010cube graph, and if <jats:italic>m<\/jats:italic> \u2265 <jats:italic>D<\/jats:italic> then <jats:italic>G<\/jats:italic> \u00d7 <jats:italic>Q<\/jats:italic><jats:sub><jats:italic>m<\/jats:italic><\/jats:sub> has a quadrilateral embedding.<\/jats:p>","DOI":"10.1002\/jgt.3190040105","type":"journal-article","created":{"date-parts":[[2007,5,29]],"date-time":"2007-05-29T06:19:35Z","timestamp":1180419575000},"page":"31-42","source":"Crossref","is-referenced-by-count":29,"title":["Genus of cartesian products of regular bipartite graphs"],"prefix":"10.1002","volume":"4","author":[{"given":"Toma\u017e","family":"Pisanski","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","first-page":"108","article-title":"On imbeddings of hypergraphs","volume":"5","author":"Alavi Y.","year":"1976","journal-title":"Graph Theory Newslett."},{"key":"e_1_2_1_3_2","doi-asserted-by":"publisher","DOI":"10.1016\/0095-8956(76)90018-6"},{"key":"e_1_2_1_4_2","volume-title":"Graphs and Digraphs","author":"Behzad M.","year":"1979"},{"key":"e_1_2_1_5_2","doi-asserted-by":"publisher","DOI":"10.4153\/CJM-1965-048-6"},{"key":"e_1_2_1_6_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-349-03521-2"},{"key":"e_1_2_1_7_2","doi-asserted-by":"publisher","DOI":"10.1016\/0095-8956(77)90068-5"},{"key":"e_1_2_1_8_2","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365X(77)90131-5"},{"key":"e_1_2_1_9_2","doi-asserted-by":"publisher","DOI":"10.21236\/AD0705364"},{"key":"e_1_2_1_10_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF02960735"},{"key":"e_1_2_1_11_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF02993245"},{"key":"e_1_2_1_12_2","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365X(76)90042-X"},{"key":"e_1_2_1_13_2","doi-asserted-by":"publisher","DOI":"10.2307\/1995503"},{"key":"e_1_2_1_14_2","doi-asserted-by":"publisher","DOI":"10.2307\/1996269"},{"key":"e_1_2_1_15_2","volume-title":"Graphs, Groups and Surfaces","author":"White A. 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