{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,18]],"date-time":"2026-02-18T09:28:17Z","timestamp":1771406897005,"version":"3.50.1"},"reference-count":21,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2025,12,10]],"date-time":"2025-12-10T00:00:00Z","timestamp":1765324800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/4.0\/"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2026,3]]},"abstract":"Abstract<\/jats:title>\n \n The famous Sidorenko\u2019s conjecture asserts that for every bipartite graph\n \n \n \n $H$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , the number of homomorphisms from\n \n \n \n $H$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n to a graph\n \n \n \n $G$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n with given edge density is minimised when\n \n \n \n $G$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n is pseudorandom. We prove that for any graph\n \n \n \n $H$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , a graph obtained from replacing edges of\n \n \n \n $H$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n by generalised theta graphs consisting of even paths satisfies Sidorenko\u2019s conjecture, provided a certain divisibility condition on the number of paths. To achieve this, we prove unconditionally that bipartite graphs obtained from replacing each edge of a complete graph with a generalised theta graph satisfy Sidorenko\u2019s conjecture, which extends a result of Conlon, Kim, Lee and Lee [J. Lond. Math. Soc., 2018].\n <\/jats:p>","DOI":"10.1017\/s0963548325100242","type":"journal-article","created":{"date-parts":[[2025,12,10]],"date-time":"2025-12-10T07:11:18Z","timestamp":1765350678000},"page":"269-279","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":0,"title":["Sidorenko\u2019s conjecture for subdivisions and theta substitutions"],"prefix":"10.1017","volume":"35","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1996-6801","authenticated-orcid":false,"given":"Seonghyuk","family":"Im","sequence":"first","affiliation":[{"name":"KAIST"},{"name":"Institute of Basic Science (IBS)"}]},{"given":"Ruonan","family":"Li","sequence":"additional","affiliation":[{"name":"Northwestern Polytechnical University"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5735-7321","authenticated-orcid":false,"given":"Hong","family":"Liu","sequence":"additional","affiliation":[{"name":"Institute of Basic Science (IBS)"}]}],"member":"56","published-online":{"date-parts":[[2025,12,10]]},"reference":[{"key":"S0963548325100242_ref4","doi-asserted-by":"publisher","DOI":"10.1007\/s00039-010-0097-0"},{"key":"S0963548325100242_ref19","volume-title":"Large Networks and Graph Limits","volume":"60","author":"Lov\u00e1sz","year":"2012"},{"key":"S0963548325100242_ref8","doi-asserted-by":"publisher","DOI":"10.1007\/s11856-010-0005-1"},{"key":"S0963548325100242_ref21","unstructured":"[21] Coregliano, L. N. and Razborov, A. A. (2021) Biregularity in Sidorenko\u2019s conjecture. arXiv:2108.06599."},{"key":"S0963548325100242_ref11","unstructured":"[11] Szegedy, B. (2015) An information theoretic approach to Sidorenko\u2019s conjecture. arXiv:1406.6738."},{"key":"S0963548325100242_ref20","unstructured":"[20] Szegedy, B. (2015) Sparse graph limits, entropy maximization and transitive graphs. arXiv:1504.00858."},{"key":"S0963548325100242_ref9","doi-asserted-by":"publisher","DOI":"10.1090\/tran\/6487"},{"key":"S0963548325100242_ref5","doi-asserted-by":"publisher","DOI":"10.1112\/jlms.12142"},{"key":"S0963548325100242_ref13","doi-asserted-by":"publisher","DOI":"10.1016\/j.jctb.2009.05.005"},{"key":"S0963548325100242_ref15","unstructured":"[15] Brada\u010d, D. , Sudakov, B. and Wigderson, Y. (2024) Counting subgraphs in locally dense graphs. arXiv:2406.12418."},{"key":"S0963548325100242_ref7","doi-asserted-by":"publisher","DOI":"10.1137\/22M1526794"},{"key":"S0963548325100242_ref17","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.20974"},{"key":"S0963548325100242_ref14","doi-asserted-by":"publisher","DOI":"10.1016\/j.jctb.2013.12.002"},{"key":"S0963548325100242_ref12","doi-asserted-by":"publisher","DOI":"10.1007\/BF02125347"},{"key":"S0963548325100242_ref2","doi-asserted-by":"publisher","DOI":"10.1007\/BF02988307"},{"key":"S0963548325100242_ref6","first-page":"2","article-title":"Sidorenko\u2019s conjecture for blow-ups","volume":"13","author":"Conlon","year":"2021","journal-title":"Discrete Anal"},{"key":"S0963548325100242_ref3","unstructured":"[3] Erd\u0151s, P. and Simonovits, M. (1984) Cube-supersaturated graphs and related problems. Progress in Graph Theory (Waterloo, Ont 1982). 203\u2013218."},{"key":"S0963548325100242_ref1","first-page":"50","article-title":"Inequalities for functionals generated by bipartite graphs","volume":"3","author":"Sidorenko","year":"1991","journal-title":"Diskret. Mat"},{"key":"S0963548325100242_ref18","unstructured":"[18] Chen, H. , Lin, Y. and Ma, J. (2024) Kohayakawa\u2013Nagle\u2013R\u00f6dl\u2013Schacht conjecture for subdivisions. arXiv:2407.10861."},{"key":"S0963548325100242_ref10","unstructured":"[10] Li, J. L. X. and Szegedy, B. (2011) On the logarithimic calculus and Sidorenko\u2019s conjecture. arXiv:1107.1153."},{"key":"S0963548325100242_ref16","doi-asserted-by":"publisher","DOI":"10.1112\/S0010437X24007681"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548325100242","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,2,18]],"date-time":"2026-02-18T08:53:50Z","timestamp":1771404830000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548325100242\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,12,10]]},"references-count":21,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2026,3]]}},"alternative-id":["S0963548325100242"],"URL":"https:\/\/doi.org\/10.1017\/s0963548325100242","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,12,10]]},"assertion":[{"value":"\u00a9 The Author(s), 2025. 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-NonCommercial-ShareAlike licence (https:\/\/creativecommons.org\/licenses\/by-nc-sa\/4.0\/), which permits non-commercial re-use, distribution, and reproduction in any medium, provided the same Creative Commons licence is used to distribute the re-used or adapted article and the original article is properly cited. The written permission of Cambridge University Press or the rights holder(s) must be obtained prior to any commercial use.","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"}]}}