{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,27]],"date-time":"2025-10-27T21:06:32Z","timestamp":1761599192306,"version":"build-2065373602"},"reference-count":52,"publisher":"Springer Science and Business Media LLC","issue":"5","license":[{"start":{"date-parts":[[2025,10,1]],"date-time":"2025-10-01T00:00:00Z","timestamp":1759276800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,10,2]],"date-time":"2025-10-02T00:00:00Z","timestamp":1759363200000},"content-version":"vor","delay-in-days":1,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"Swiss Federal Institute of Technology Zurich"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Combinatorica"],"published-print":{"date-parts":[[2025,10]]},"abstract":"Abstract<\/jats:title>\n \n In a recent work, Allen, B\u00f6ttcher, H\u00e0n, Kohayakawa, and Person provided a first general analogue of the blow-up lemma applicable to sparse (pseudo)random graphs thus generalising the classic tool of Koml\u00f3s, S\u00e1rk\u00f6zy, and Szemer\u00e9di. Roughly speaking, they showed that with high probability in the random graph\n \n \n $$G_{n, p}$$<\/jats:tex-math>\n \n \n G<\/mml:mi>\n \n n<\/mml:mi>\n ,<\/mml:mo>\n p<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n for\n \n \n $$p \\geqslant C(\\log n\/n)^{1\/\\Delta }$$<\/jats:tex-math>\n \n \n p<\/mml:mi>\n \u2a7e<\/mml:mo>\n C<\/mml:mi>\n \n \n (<\/mml:mo>\n log<\/mml:mo>\n n<\/mml:mi>\n \/<\/mml:mo>\n n<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \n 1<\/mml:mn>\n \/<\/mml:mo>\n \u0394<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , sparse regular pairs behave similarly as complete bipartite graphs with respect to embedding a spanning graph\n H<\/jats:italic>\n with\n \n \n $$\\Delta (H) \\leqslant \\Delta$$<\/jats:tex-math>\n \n \n \u0394<\/mml:mi>\n (<\/mml:mo>\n H<\/mml:mi>\n )<\/mml:mo>\n \u2a7d<\/mml:mo>\n \u0394<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . However, this is typically only optimal when\n \n \n $$\\Delta \\in \\{2,3\\}$$<\/jats:tex-math>\n \n \n \u0394<\/mml:mi>\n \u2208<\/mml:mo>\n {<\/mml:mo>\n 2<\/mml:mn>\n ,<\/mml:mo>\n 3<\/mml:mn>\n }<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n and\n H<\/jats:italic>\n either contains a triangle (\n \n \n $$\\Delta = 2$$<\/jats:tex-math>\n \n \n \u0394<\/mml:mi>\n =<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n ) or many copies of\n \n \n $$K_4$$<\/jats:tex-math>\n \n \n K<\/mml:mi>\n 4<\/mml:mn>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n (\n \n \n $$\\Delta = 3$$<\/jats:tex-math>\n \n \n \u0394<\/mml:mi>\n =<\/mml:mo>\n 3<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n ). We go beyond this barrier for the first time and present a sparse blow-up lemma for cycles\n \n \n $$C_{2k-1}, C_{2k}$$<\/jats:tex-math>\n \n \n \n C<\/mml:mi>\n \n 2<\/mml:mn>\n k<\/mml:mi>\n -<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n ,<\/mml:mo>\n \n C<\/mml:mi>\n \n 2<\/mml:mn>\n k<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , for all\n \n \n $$k \\geqslant 2$$<\/jats:tex-math>\n \n \n k<\/mml:mi>\n \u2a7e<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , and densities\n \n \n $$p \\geqslant Cn^{-(k-1)\/k}$$<\/jats:tex-math>\n \n \n p<\/mml:mi>\n \u2a7e<\/mml:mo>\n C<\/mml:mi>\n \n n<\/mml:mi>\n \n -<\/mml:mo>\n (<\/mml:mo>\n k<\/mml:mi>\n -<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n \/<\/mml:mo>\n k<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , which is in a way best possible. As an application of our blow-up lemma we fully resolve a question of Nenadov and \u0160kori\u0107 regarding resilience of cycle factors in sparse random graphs.\n <\/jats:p>","DOI":"10.1007\/s00493-025-00171-1","type":"journal-article","created":{"date-parts":[[2025,10,2]],"date-time":"2025-10-02T15:41:22Z","timestamp":1759419682000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Blow-up Lemma for Cycles in Sparse Random Graphs"],"prefix":"10.1007","volume":"45","author":[{"given":"Milo\u0161","family":"Truji\u0107","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,10,2]]},"reference":[{"key":"171_CR1","doi-asserted-by":"crossref","unstructured":"Allen, P., B\u00f6ttcher, J., Ehrenm\u00fcller, J., Taraz, A.: The bandwidth theorem in sparse graphs. Adv. Comb., 60, 2020. 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