{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,3]],"date-time":"2022-04-03T05:32:38Z","timestamp":1648963958436},"reference-count":8,"publisher":"Cambridge University Press (CUP)","issue":"6","license":[{"start":{"date-parts":[[2007,11,1]],"date-time":"2007-11-01T00:00:00Z","timestamp":1193875200000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2007,11]]},"abstract":"<jats:p>In 1972, Rosenfeld asked if every triangle-free graph could be embedded in the unit sphere <jats:italic>S<\/jats:italic><jats:sup><jats:italic>d<\/jats:italic><\/jats:sup> in such a way that two vertices joined by an edge have distance more than <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0963548307008528_inline1\"><jats:alt-text>$\\sqrt 3$<\/jats:alt-text><\/jats:inline-graphic>(ie, distance more than 2\u03c0\/3 on the sphere). In 1978, Larman [LAR] disproved this conjecture, constructing a triangle-free graph for which the minimum length of an edge could not exceed <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0963548307008528_inline2\"><jats:alt-text>$\\sqrt{8\/3}$<\/jats:alt-text><\/jats:inline-graphic>. In addition, he conjectured that the right answer would be <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0963548307008528_inline3\"><jats:alt-text>$\\sqrt{2}$<\/jats:alt-text><\/jats:inline-graphic>, which is not better than the class of all graphs. Larman'sconjecture was independently proved by Rosenfeld [MR] and R\u00f6dl [VR[. In this last paper it was shown that no bound better than <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0963548307008528_inline4\"><jats:alt-text>$\\sqrt 2$<\/jats:alt-text><\/jats:inline-graphic> can be found for graphs with arbitrarily large odd girth. We prove in this paper that this is stilltrue for arbitrarily large girth. We discuss then the case of triangle-free graphs with linear minimum degree.<\/jats:p>","DOI":"10.1017\/s0963548307008528","type":"journal-article","created":{"date-parts":[[2007,7,20]],"date-time":"2007-07-20T08:18:56Z","timestamp":1184919536000},"page":"829-832","source":"Crossref","is-referenced-by-count":0,"title":["Graphs with Large Girth Not Embeddable in the Sphere"],"prefix":"10.1017","volume":"16","author":[{"given":"PIERRE","family":"CHARBIT","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"ST\u00c9PHAN","family":"THOMASS\u00c9","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2007,11,1]]},"reference":[{"key":"S0963548307008528_ref4","doi-asserted-by":"publisher","DOI":"10.1145\/274787.274791"},{"key":"S0963548307008528_ref1","unstructured":"[1] Brandt S. and Thomass\u00e9 S. Dense triangle-free graphs are four-colorable: A solution to the ErdH\u0151s\u2013Simonovits problem. to appear in J. Combin Theory Ser. B."},{"key":"S0963548307008528_ref5","doi-asserted-by":"publisher","DOI":"10.1016\/0097-3165(78)90004-3"},{"key":"S0963548307008528_ref8","doi-asserted-by":"publisher","DOI":"10.1016\/0095-8956(82)90069-7"},{"key":"S0963548307008528_ref7","doi-asserted-by":"publisher","DOI":"10.1007\/BF02579146"},{"key":"S0963548307008528_ref3","doi-asserted-by":"publisher","DOI":"10.1145\/227683.227684"},{"key":"S0963548307008528_ref2","first-page":"153","volume-title":"Graph Theory and Related Topics","author":"ErdH\u0151s","year":"1979"},{"key":"S0963548307008528_ref6","first-page":"143","article-title":"Embedding graphs in Euclidean spaces: An exploration guided by Paul Erdos","volume":"6","author":"Ne\u011aset\u011aril","year":"1997","journal-title":"Geombinatorics"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548307008528","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,3,29]],"date-time":"2019-03-29T19:14:15Z","timestamp":1553886855000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548307008528\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,11]]},"references-count":8,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2007,11]]}},"alternative-id":["S0963548307008528"],"URL":"https:\/\/doi.org\/10.1017\/s0963548307008528","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2007,11]]}}}