{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,12]],"date-time":"2026-03-12T13:34:05Z","timestamp":1773322445745,"version":"3.50.1"},"reference-count":15,"publisher":"Cambridge University Press (CUP)","issue":"1-2","license":[{"start":{"date-parts":[[2012,3,19]],"date-time":"2012-03-19T00:00:00Z","timestamp":1332115200000},"content-version":"unspecified","delay-in-days":18,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2012,3]]},"abstract":"<jats:p>Li, Nikiforov and Schelp [13] conjectured that any 2-edge coloured graph <jats:italic>G<\/jats:italic> with order <jats:italic>n<\/jats:italic> and minimum degree \u03b4(<jats:italic>G<\/jats:italic>) &gt; 3<jats:italic>n<\/jats:italic>\/4 contains a monochromatic cycle of length \u2113, for all \u2113 \u2208 [4, \u2308<jats:italic>n<\/jats:italic>\/2\u2309]. We prove this conjecture for sufficiently large <jats:italic>n<\/jats:italic> and also find all 2-edge coloured graphs with \u03b4(<jats:italic>G<\/jats:italic>)=3<jats:italic>n<\/jats:italic>\/4 that do not contain all such cycles. Finally, we show that, for all \u03b4&gt;0 and <jats:italic>n<\/jats:italic>&gt;<jats:italic>n<\/jats:italic><jats:sub>0<\/jats:sub>(\u03b4), if <jats:italic>G<\/jats:italic> is a 2-edge coloured graph of order <jats:italic>n<\/jats:italic> with \u03b4(<jats:italic>G<\/jats:italic>) \u2265 3<jats:italic>n<\/jats:italic>\/4, then one colour class either contains a monochromatic cycle of length at least (2\/3+\u03b4\/2)<jats:italic>n<\/jats:italic>, or contains monochromatic cycles of all lengths \u2113 \u2208 [3, (2\/3\u2212\u03b4)<jats:italic>n<\/jats:italic>].<\/jats:p>","DOI":"10.1017\/s0963548312000090","type":"journal-article","created":{"date-parts":[[2012,3,19]],"date-time":"2012-03-19T15:20:59Z","timestamp":1332170459000},"page":"57-87","source":"Crossref","is-referenced-by-count":25,"title":["Monochromatic Cycles in 2-Coloured Graphs"],"prefix":"10.1017","volume":"21","author":[{"given":"F. S.","family":"BENEVIDES","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"T.","family":"\u0141UCZAK","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"A.","family":"SCOTT","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"J.","family":"SKOKAN","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"M.","family":"WHITE","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2012,3,19]]},"reference":[{"key":"S0963548312000090_ref4","unstructured":"[4] Bollob\u00e1s B. , Benevides F. , \u0141uczak T. and Skokan J. (2011) Graphs with large minimum degrees arrow even cycles. Manuscript."},{"key":"S0963548312000090_ref9","doi-asserted-by":"publisher","DOI":"10.1007\/BF02024498"},{"key":"S0963548312000090_ref7","doi-asserted-by":"publisher","DOI":"10.1016\/0095-8956(72)90020-2"},{"key":"S0963548312000090_ref14","first-page":"399","article-title":"Regular partitions of graphs.","volume":"260","author":"Szemer\u00e9di","year":"1976","journal-title":"Colloques Internationaux CNRS: Probl\u00e8mes Combinatoires et Th\u00e9orie des Graphes"},{"key":"S0963548312000090_ref12","first-page":"295","volume-title":"Combinatorics: Paul Erd\u0151s is Eighty","author":"Koml\u00f3s","year":"1996"},{"key":"S0963548312000090_ref13","doi-asserted-by":"publisher","DOI":"10.1016\/j.disc.2010.09.009"},{"key":"S0963548312000090_ref6","doi-asserted-by":"publisher","DOI":"10.1016\/0095-8956(74)90052-5"},{"key":"S0963548312000090_ref8","doi-asserted-by":"publisher","DOI":"10.1112\/plms\/s3-2.1.69"},{"key":"S0963548312000090_ref2","first-page":"258","article-title":"Sur le couplage maximum d'un graphe (in French).","volume":"247","author":"Berge","year":"1958","journal-title":"CR Acad. Sci. Paris"},{"key":"S0963548312000090_ref1","doi-asserted-by":"publisher","DOI":"10.1016\/j.jctb.2008.12.002"},{"key":"S0963548312000090_ref15","doi-asserted-by":"publisher","DOI":"10.1112\/jlms\/s1-22.2.107"},{"key":"S0963548312000090_ref10","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548311000599"},{"key":"S0963548312000090_ref11","doi-asserted-by":"publisher","DOI":"10.1016\/S0012-365X(03)00076-1"},{"key":"S0963548312000090_ref5","doi-asserted-by":"publisher","DOI":"10.1016\/0095-8956(71)90016-5"},{"key":"S0963548312000090_ref3","volume-title":"Extremal Graph Theory","author":"Bollob\u00e1s","year":"1978"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548312000090","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,26]],"date-time":"2019-04-26T01:25:04Z","timestamp":1556241904000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548312000090\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,3]]},"references-count":15,"journal-issue":{"issue":"1-2","published-print":{"date-parts":[[2012,3]]}},"alternative-id":["S0963548312000090"],"URL":"https:\/\/doi.org\/10.1017\/s0963548312000090","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012,3]]}}}