{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,4,25]],"date-time":"2023-04-25T04:32:47Z","timestamp":1682397167217},"reference-count":9,"publisher":"Cambridge University Press (CUP)","license":[{"start":{"date-parts":[[2023,4,24]],"date-time":"2023-04-24T00:00:00Z","timestamp":1682294400000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["Combinator. Probab. Comp."],"abstract":"Abstract<\/jats:title>\n\t A random two-cell embedding of a given graph \n\t \n\t\t\n\t\t\n$G$\n<\/jats:tex-math>\n\t <\/jats:alternatives>\n\t <\/jats:inline-formula> is obtained by choosing a random local rotation around every vertex. We analyse the expected number of faces of such an embedding, which is equivalent to studying its average genus. In 1991, Stahl [5] proved that the expected number of faces in a random embedding of an arbitrary graph of order \n\t \n\t\t\n\t\t\n$n$\n<\/jats:tex-math>\n\t <\/jats:alternatives>\n\t <\/jats:inline-formula> is at most \n\t \n\t\t\n\t\t\n$n\\log (n)$\n<\/jats:tex-math>\n\t <\/jats:alternatives>\n\t <\/jats:inline-formula>. While there are many families of graphs whose expected number of faces is \n\t \n\t\t\n\t\t\n$\\Theta (n)$\n<\/jats:tex-math>\n\t <\/jats:alternatives>\n\t <\/jats:inline-formula>, none are known where the expected number would be super-linear. This led the authors of [1] to conjecture that there is a linear upper bound. In this note we confirm their conjecture by proving that for any \n\t \n\t\t\n\t\t\n$n$\n<\/jats:tex-math>\n\t <\/jats:alternatives>\n\t <\/jats:inline-formula>-vertex multigraph, the expected number of faces in a random two-cell embedding is at most \n\t \n\t\t\n\t\t\n$2n\\log (2\\mu )$\n<\/jats:tex-math>\n\t <\/jats:alternatives>\n\t <\/jats:inline-formula>, where \n\t \n\t\t\n\t\t\n$\\mu$\n<\/jats:tex-math>\n\t <\/jats:alternatives>\n\t <\/jats:inline-formula> is the maximum edge-multiplicity. This bound is best possible up to a constant factor.<\/jats:p>","DOI":"10.1017\/s096354832300010x","type":"journal-article","created":{"date-parts":[[2023,4,24]],"date-time":"2023-04-24T09:25:57Z","timestamp":1682328357000},"page":"1-9","update-policy":"http:\/\/dx.doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":0,"title":["Expected number of faces in a random embedding of any graph is at most linear"],"prefix":"10.1017","author":[{"given":"Jesse","family":"Campion Loth","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Bojan","family":"Mohar","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2023,4,24]]},"reference":[{"key":"S096354832300010X_ref1","doi-asserted-by":"publisher","DOI":"10.1090\/proc\/15899"},{"key":"S096354832300010X_ref5","doi-asserted-by":"publisher","DOI":"10.1016\/0095-8956(91)90063-P"},{"key":"S096354832300010X_ref4","doi-asserted-by":"publisher","DOI":"10.56021\/9780801866890"},{"key":"S096354832300010X_ref8","volume-title":"Graphs, Groups and Surfaces","volume":"8","author":"White","year":"1973"},{"key":"S096354832300010X_ref9","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548300001395"},{"key":"S096354832300010X_ref3","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-38361-1"},{"key":"S096354832300010X_ref6","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.3190200102"},{"key":"S096354832300010X_ref2","volume-title":"Topological Graph Theory","author":"Gross","year":"1987"},{"key":"S096354832300010X_ref7","doi-asserted-by":"publisher","DOI":"10.1016\/j.ejc.2011.01.011"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S096354832300010X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,4,24]],"date-time":"2023-04-24T09:26:02Z","timestamp":1682328362000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S096354832300010X\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,4,24]]},"references-count":9,"alternative-id":["S096354832300010X"],"URL":"https:\/\/doi.org\/10.1017\/s096354832300010x","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,4,24]]},"assertion":[{"value":"\u00a9 The Author(s), 2023. Published by Cambridge University Press","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}},{"value":"This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https:\/\/creativecommons.org\/licenses\/by\/4.0\/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.","name":"license","label":"License","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}},{"value":"This content has been made available to all.","name":"free","label":"Free to read"}]}}