{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,2]],"date-time":"2026-06-02T07:48:04Z","timestamp":1780386484368,"version":"3.54.1"},"reference-count":9,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2024,5,2]],"date-time":"2024-05-02T00:00:00Z","timestamp":1714608000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2024,5,2]],"date-time":"2024-05-02T00:00:00Z","timestamp":1714608000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"Korea Advanced Institute of Science and Technology"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Combinatorica"],"published-print":{"date-parts":[[2024,8]]},"abstract":"Abstract<\/jats:title>We prove that every properly edge-colored n<\/jats:italic>-vertex graph with average degree at least $$32(\\log 5n)^2$$<\/jats:tex-math>\n \n 32<\/mml:mn>\n \n \n (<\/mml:mo>\n log<\/mml:mo>\n 5<\/mml:mn>\n n<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n 2<\/mml:mn>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> contains a rainbow cycle, improving upon the $$(\\log n)^{2+o(1)}$$<\/jats:tex-math>\n \n \n (<\/mml:mo>\n log<\/mml:mo>\n n<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \n 2<\/mml:mn>\n +<\/mml:mo>\n o<\/mml:mi>\n (<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> bound due to Tomon. We also prove that every properly edge-colored n<\/jats:italic>-vertex graph with at least $$10^5 k^3 n^{1+1\/k}$$<\/jats:tex-math>\n \n \n 10<\/mml:mn>\n 5<\/mml:mn>\n <\/mml:msup>\n \n k<\/mml:mi>\n 3<\/mml:mn>\n <\/mml:msup>\n \n n<\/mml:mi>\n \n 1<\/mml:mn>\n +<\/mml:mo>\n 1<\/mml:mn>\n \/<\/mml:mo>\n k<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> edges contains a rainbow 2k<\/jats:italic>-cycle, which improves the previous bound $$2^{ck^2}n^{1+1\/k}$$<\/jats:tex-math>\n \n \n 2<\/mml:mn>\n \n c<\/mml:mi>\n \n k<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:msup>\n \n n<\/mml:mi>\n \n 1<\/mml:mn>\n +<\/mml:mo>\n 1<\/mml:mn>\n \/<\/mml:mo>\n k<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> obtained by Janzer. Our method using homomorphism inequalities and a lopsided regularization lemma also provides a simple way to prove the Erd\u0151s\u2013Simonovits supersaturation theorem for even cycles, which may be of independent interest.<\/jats:p>","DOI":"10.1007\/s00493-024-00101-7","type":"journal-article","created":{"date-parts":[[2024,5,2]],"date-time":"2024-05-02T10:01:45Z","timestamp":1714644105000},"page":"909-919","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Rainbow Cycles in Properly Edge-Colored Graphs"],"prefix":"10.1007","volume":"44","author":[{"given":"Jaehoon","family":"Kim","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Joonkyung","family":"Lee","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hong","family":"Liu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Tuan","family":"Tran","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2024,5,2]]},"reference":[{"key":"101_CR1","unstructured":"Alon, N.,Buci\u0107, M., Sauermann, L., Zakharov, D., Zamir, O.: Essentially tight bounds for rainbow cycles in proper edge-colourings. arXiv:2309.04460"},{"key":"101_CR2","doi-asserted-by":"publisher","first-page":"97","DOI":"10.1016\/0095-8956(74)90052-5","volume":"16","author":"JA Bondy","year":"1974","unstructured":"Bondy, J.A., Simonovits, M.: Cycles of even length in graphs. J. Combin. Theory Ser. B 16, 97\u2013105 (1974)","journal-title":"J. Combin. Theory Ser. B"},{"issue":"5","key":"101_CR3","doi-asserted-by":"publisher","first-page":"905","DOI":"10.1016\/j.ejc.2013.01.003","volume":"34","author":"S Das","year":"2013","unstructured":"Das, S., Lee, C., Sudakov, B.: Rainbow Tur\u00e1n problem for even cycles. Eur. J. Combin. 34(5), 905\u2013915 (2013)","journal-title":"Eur. J. Combin."},{"key":"101_CR4","unstructured":"Erd\u0151s, P., and Simonovits, M.: Cube-supersaturated graphs and related problems. In: Progress in Graph Theory, pp.\u00a0203\u2013218. Academic Press, Toronto (1984)"},{"issue":"2","key":"101_CR5","doi-asserted-by":"publisher","first-page":"813","DOI":"10.1007\/s11856-022-2380-9","volume":"253","author":"O Janzer","year":"2023","unstructured":"Janzer, O.: Rainbow Tur\u00e1n number of even cycles, repeated patterns and blow-ups of cycles. Israel J. Math. 253(2), 813\u2013840 (2023)","journal-title":"Israel J. Math."},{"key":"101_CR6","doi-asserted-by":"crossref","unstructured":"Janzer, O., Sudakov, B.: On the Tur\u00e0in number of the hypercube. Forum Math. Sigma 12, e38 (2024)","DOI":"10.1017\/fms.2024.27"},{"key":"101_CR7","doi-asserted-by":"publisher","first-page":"1238","DOI":"10.1137\/100819254","volume":"26","author":"T Jiang","year":"2012","unstructured":"Jiang, T., Seiver, R.: Tur\u00e1n numbers of subdivided graphs. SIAM J. Discrete Math. 26, 1238\u20131255 (2012)","journal-title":"SIAM J. Discrete Math."},{"issue":"1","key":"101_CR8","doi-asserted-by":"publisher","first-page":"109","DOI":"10.1017\/S0963548306007760","volume":"16","author":"P Keevash","year":"2007","unstructured":"Keevash, P., Mubayi, D., Sudakov, B., Verstra\u00ebte, J.: Rainbow Tur\u00e1n problems. Combin. Probab. Comput. 16(1), 109\u2013126 (2007)","journal-title":"Combin. Probab. Comput."},{"key":"101_CR9","doi-asserted-by":"publisher","unstructured":"Tomon, I.: Robust (rainbow) subdivisions and simplicial cycles. Advances in Combinatorics. https:\/\/doi.org\/10.19086\/aic.2024.1 (2024)","DOI":"10.19086\/aic.2024.1"}],"container-title":["Combinatorica"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00493-024-00101-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00493-024-00101-7\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00493-024-00101-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,7,24]],"date-time":"2024-07-24T17:05:59Z","timestamp":1721840759000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00493-024-00101-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,5,2]]},"references-count":9,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2024,8]]}},"alternative-id":["101"],"URL":"https:\/\/doi.org\/10.1007\/s00493-024-00101-7","relation":{},"ISSN":["0209-9683","1439-6912"],"issn-type":[{"value":"0209-9683","type":"print"},{"value":"1439-6912","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,5,2]]},"assertion":[{"value":"6 November 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"25 January 2024","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"25 February 2024","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"2 May 2024","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}