{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,10,22]],"date-time":"2023-10-22T12:12:14Z","timestamp":1697976734687},"reference-count":10,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2006,10,5]],"date-time":"2006-10-05T00:00:00Z","timestamp":1160006400000},"content-version":"vor","delay-in-days":6427,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Graph Theory"],"published-print":{"date-parts":[[1989,3]]},"abstract":"Abstract<\/jats:title>We show that the non\u2010commutative semidirect product \u0393 of \u21249<\/jats:sub> by \u21243<\/jats:sub> has orientable genus 4. In other words, some Cayley graph of \u0393 embeds in an orientable surface of genus 4 (Euler characteristic \u22126), but no Cayley graph of \u0393 embeds in an orientable surface of genus less than 4 (Euler characteristic greater than \u22126). We also show that some Cayley graph of \u0393 embeds in a (non\u2010orientable) surface of Euler characteristic \u22123, but no Cayley graph of \u0393 embeds in a surface of Euler characteristic greater than \u22123. \u0393 is the first known example of a group whose orientable Euler characteristic and non\u2010orientable Euler characteristic differ by more than 1. Our results also complete the determination of the orientable genus of each group of order less than 32.<\/jats:p>","DOI":"10.1002\/jgt.3190130108","type":"journal-article","created":{"date-parts":[[2007,5,26]],"date-time":"2007-05-26T12:18:31Z","timestamp":1180181911000},"page":"49-61","source":"Crossref","is-referenced-by-count":5,"title":["On the genus of the semidirect product of \u21249<\/sub> by \u21243<\/sub>"],"prefix":"10.1002","volume":"13","author":[{"given":"Matthew G.","family":"Brin","sequence":"first","affiliation":[]},{"given":"David E.","family":"Rauschenberg","sequence":"additional","affiliation":[]},{"given":"Craig C.","family":"Squier","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2006,10,5]]},"reference":[{"key":"e_1_2_1_2_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0195-6698(88)80002-7"},{"key":"e_1_2_1_3_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-21946-1"},{"key":"e_1_2_1_4_2","volume-title":"Topological Graph Theory","author":"Gross J. L.","year":"1987"},{"key":"e_1_2_1_5_2","volume-title":"The Theory of Groups","author":"Hall M.","year":"1959"},{"key":"e_1_2_1_6_2","unstructured":"V. K.Proulx Classification of the toroidal groups Ph.D. dissertation Columbia University (1977)."},{"key":"e_1_2_1_7_2","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.3190050316"},{"key":"e_1_2_1_8_2","doi-asserted-by":"publisher","DOI":"10.1016\/0095-8956(83)90009-6"},{"key":"e_1_2_1_9_2","doi-asserted-by":"publisher","DOI":"10.1016\/0095-8956(84)90031-5"},{"key":"e_1_2_1_10_2","volume-title":"Graphs, Groups, and Surfaces","author":"White A. T.","year":"1973"},{"key":"e_1_2_1_11_2","unstructured":"A. T.White The genus parameter for groups preprint."}],"container-title":["Journal of Graph Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fjgt.3190130108","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/jgt.3190130108","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,21]],"date-time":"2023-10-21T16:52:35Z","timestamp":1697907155000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/jgt.3190130108"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1989,3]]},"references-count":10,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1989,3]]}},"alternative-id":["10.1002\/jgt.3190130108"],"URL":"https:\/\/doi.org\/10.1002\/jgt.3190130108","archive":["Portico"],"relation":{},"ISSN":["0364-9024","1097-0118"],"issn-type":[{"value":"0364-9024","type":"print"},{"value":"1097-0118","type":"electronic"}],"subject":[],"published":{"date-parts":[[1989,3]]}}}