[{"publisher":"Springer Nature","has_accepted_license":"1","acknowledgement":"We would like to thank the referees for a number of helpful comments and suggestions, which have substantially improved the paper. Open access funding provided by Institute of Science and Technology (IST Austria).","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","ddc":["510"],"oa":1,"file":[{"relation":"main_file","access_level":"open_access","checksum":"47b0031d90b0e6b9a843f422a1486089","content_type":"application/pdf","file_id":"21228","file_size":539646,"file_name":"2026_Combinatorica_Kwan.pdf","success":1,"date_created":"2026-02-16T09:52:38Z","date_updated":"2026-02-16T09:52:38Z","creator":"dernst"}],"corr_author":"1","abstract":[{"lang":"eng","text":"One of the foundational theorems of extremal graph theory is Dirac’s theorem, which\r\nsays that if an n-vertex graph G has minimum degree at least n/2, then G has a\r\nHamilton cycle, and therefore a perfect matching (if n is even). Later work by Sárközy,\r\nSelkow and Szemerédi showed that in fact Dirac graphs have many Hamilton cycles\r\nand perfect matchings, culminating in a result of Cuckler and Kahn that gives a precise\r\ndescription of the numbers of Hamilton cycles and perfect matchings in a Dirac graph\r\nG (in terms of an entropy-like parameter of G). In this paper we extend Cuckler\r\nand Kahn’s result to perfect matchings in hypergraphs. For positive integers d < k,\r\nand for n divisible by k, let md (k, n) be the minimum d-degree that ensures the\r\nexistence of a perfect matching in an n-vertex k-uniform hypergraph. In general, it is\r\nan open question to determine (even asymptotically) the values of md (k, n), but we are\r\nnonetheless able to prove an analogue of the Cuckler–Kahn theorem, showing that if\r\nan n-vertex k-uniform hypergraph G has minimum d-degree at least (1+γ )md (k, n)\r\n(for any constantγ > 0), then the number of perfect matchings in G is controlled by\r\nan entropy-like parameter of G. This strengthens cruder estimates arising from work\r\nof Kang–Kelly–Kühn–Osthus–Pfenninger and Pham–Sah–Sawhney–Simkin."}],"language":[{"iso":"eng"}],"oa_version":"Published Version","year":"2026","volume":46,"external_id":{"arxiv":["2408.09589"]},"department":[{"_id":"MaKw"},{"_id":"MoHe"}],"intvolume":" 46","_id":"21159","article_processing_charge":"Yes (via OA deal)","date_updated":"2026-02-16T09:55:17Z","status":"public","citation":{"ieee":"M. A. Kwan, R. Safavi Hemami, and Y. Wang, “Counting perfect matchings in Dirac hypergraphs,” Combinatorica, vol. 46. Springer Nature, 2026.","mla":"Kwan, Matthew Alan, et al. “Counting Perfect Matchings in Dirac Hypergraphs.” Combinatorica, vol. 46, 5, Springer Nature, 2026, doi:10.1007/s00493-025-00194-8.","ista":"Kwan MA, Safavi Hemami R, Wang Y. 2026. Counting perfect matchings in Dirac hypergraphs. Combinatorica. 46, 5.","ama":"Kwan MA, Safavi Hemami R, Wang Y. Counting perfect matchings in Dirac hypergraphs. Combinatorica. 2026;46. doi:10.1007/s00493-025-00194-8","apa":"Kwan, M. A., Safavi Hemami, R., & Wang, Y. (2026). Counting perfect matchings in Dirac hypergraphs. Combinatorica. Springer Nature. https://doi.org/10.1007/s00493-025-00194-8","short":"M.A. Kwan, R. Safavi Hemami, Y. Wang, Combinatorica 46 (2026).","chicago":"Kwan, Matthew Alan, Roodabeh Safavi Hemami, and Yiting Wang. “Counting Perfect Matchings in Dirac Hypergraphs.” Combinatorica. Springer Nature, 2026. https://doi.org/10.1007/s00493-025-00194-8."},"publication_identifier":{"eissn":["1439-6912"],"issn":["0209-9683"]},"arxiv":1,"doi":"10.1007/s00493-025-00194-8","title":"Counting perfect matchings in Dirac hypergraphs","publication":"Combinatorica","quality_controlled":"1","OA_type":"hybrid","day":"01","fulldoi":"https://doi.org/10.1007/s00493-025-00194-8","file_date_updated":"2026-02-16T09:52:38Z","publication_status":"published","article_type":"original","date_published":"2026-02-01T00:00:00Z","scopus_import":"1","PlanS_conform":"1","OA_place":"publisher","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)"},"type":"journal_article","author":[{"full_name":"Kwan, Matthew Alan","id":"5fca0887-a1db-11eb-95d1-ca9d5e0453b3","last_name":"Kwan","first_name":"Matthew Alan","orcid":"0000-0002-4003-7567"},{"first_name":"Roodabeh","id":"72ed2640-8972-11ed-ae7b-f9c81ec75154","last_name":"Safavi Hemami","full_name":"Safavi Hemami, Roodabeh"},{"full_name":"Wang, Yiting","last_name":"Wang","id":"1917d194-076e-11ed-97cd-837255f88785","orcid":"0000-0002-2856-767X","first_name":"Yiting"}],"month":"02","article_number":"5","date_created":"2026-02-08T23:02:49Z"},{"year":"2026","volume":33,"external_id":{"arxiv":["2306.14648"]},"language":[{"iso":"eng"}],"oa_version":"Published Version","file":[{"file_name":"2026_ElectrJournCombinatorics_Morawski.pdf","file_size":399969,"success":1,"date_created":"2026-05-18T08:46:26Z","date_updated":"2026-05-18T08:46:26Z","creator":"dernst","relation":"main_file","access_level":"open_access","checksum":"9e8402cb2e8870ba7ded9ae7b308201a","content_type":"application/pdf","file_id":"21893"}],"corr_author":"1","abstract":[{"lang":"eng","text":"We show that a randomly perturbed digraph, where we start with a dense digraph Dα and add a small number of random edges to it, will typically contain a fixed orientation of a bounded-degree spanning tree. This answers a question posed by Araujo, Balogh, Krueger, Piga and Treglown and generalizes the corresponding result for randomly perturbed graphs by Krivelevich, Kwan and Sudakov. More specifically, we prove that there exists a constant c=c(α,Δ) such that if \r\nT is an oriented tree with maximum degree Δ and Dα is an n-vertex digraph with minimum semidegree αn, then the graph obtained by adding cn uniformly random edges to Dα will contain T with high probability."}],"project":[{"_id":"fc2ed2f7-9c52-11eb-aca3-c01059dda49c","grant_number":"101034413","name":"IST-BRIDGE: International postdoctoral program","call_identifier":"H2020"}],"article_processing_charge":"Yes","status":"public","date_updated":"2026-05-18T08:50:18Z","_id":"21884","intvolume":" 33","department":[{"_id":"MaKw"}],"has_accepted_license":"1","publisher":"Electronic Journal of Combinatorics","ddc":["510"],"oa":1,"acknowledgement":"We thank the anonymous referees for many helpful comments on an earlier version of this\r\narticle. Kalina Petrova was supported by grant no. CRSII5 173721 of the Swiss National\r\nScience Foundation, and by the European Union’s Horizon 2020 research and innovation\r\nprogramme under the Marie Sk lodowska-Curie grant agreement No. 101034413","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","OA_place":"publisher","date_published":"2026-05-08T00:00:00Z","article_type":"original","publication_status":"published","file_date_updated":"2026-05-18T08:46:26Z","scopus_import":"1","OA_type":"gold","day":"08","fulldoi":"https://doi.org/10.37236/13316","date_created":"2026-05-17T22:02:11Z","author":[{"full_name":"Morawski, Patryk","last_name":"Morawski","first_name":"Patryk"},{"id":"554ff4e4-f325-11ee-b0c4-a10dbd523381","last_name":"Petrova","first_name":"Kalina H","full_name":"Petrova, Kalina H"}],"month":"05","article_number":"P2.24","ec_funded":1,"type":"journal_article","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by-nd/4.0/legalcode","short":"CC BY-ND (4.0)","name":"Creative Commons Attribution-NoDerivatives 4.0 International (CC BY-ND 4.0)","image":"/image/cc_by_nd.png"},"publication":"Electronic Journal of Combinatorics","DOAJ_listed":"1","title":"Randomly perturbed digraphs also have bounded-degree spanning trees","doi":"10.37236/13316","citation":{"ieee":"P. Morawski and K. H. Petrova, “Randomly perturbed digraphs also have bounded-degree spanning trees,” Electronic Journal of Combinatorics, vol. 33, no. 2. Electronic Journal of Combinatorics, 2026.","mla":"Morawski, Patryk, and Kalina H. Petrova. “Randomly Perturbed Digraphs Also Have Bounded-Degree Spanning Trees.” Electronic Journal of Combinatorics, vol. 33, no. 2, P2.24, Electronic Journal of Combinatorics, 2026, doi:10.37236/13316.","apa":"Morawski, P., & Petrova, K. H. (2026). Randomly perturbed digraphs also have bounded-degree spanning trees. Electronic Journal of Combinatorics. Electronic Journal of Combinatorics. https://doi.org/10.37236/13316","ista":"Morawski P, Petrova KH. 2026. Randomly perturbed digraphs also have bounded-degree spanning trees. Electronic Journal of Combinatorics. 33(2), P2.24.","ama":"Morawski P, Petrova KH. Randomly perturbed digraphs also have bounded-degree spanning trees. Electronic Journal of Combinatorics. 2026;33(2). doi:10.37236/13316","short":"P. Morawski, K.H. Petrova, Electronic Journal of Combinatorics 33 (2026).","chicago":"Morawski, Patryk, and Kalina H Petrova. “Randomly Perturbed Digraphs Also Have Bounded-Degree Spanning Trees.” Electronic Journal of Combinatorics. Electronic Journal of Combinatorics, 2026. https://doi.org/10.37236/13316."},"publication_identifier":{"eissn":["1077-8926"]},"arxiv":1,"quality_controlled":"1","issue":"2"},{"arxiv":1,"citation":{"ieee":"M. Christoph, R. Nenadov, and K. H. Petrova, “The Hamilton space of pseudorandom graphs,” Journal of Combinatorial Theory Series B, vol. 176. Elsevier, pp. 254–267, 2026.","mla":"Christoph, Micha, et al. “The Hamilton Space of Pseudorandom Graphs.” Journal of Combinatorial Theory Series B, vol. 176, Elsevier, 2026, pp. 254–67, doi:10.1016/j.jctb.2025.09.002.","apa":"Christoph, M., Nenadov, R., & Petrova, K. H. (2026). The Hamilton space of pseudorandom graphs. Journal of Combinatorial Theory Series B. Elsevier. https://doi.org/10.1016/j.jctb.2025.09.002","ista":"Christoph M, Nenadov R, Petrova KH. 2026. The Hamilton space of pseudorandom graphs. Journal of Combinatorial Theory Series B. 176, 254–267.","ama":"Christoph M, Nenadov R, Petrova KH. The Hamilton space of pseudorandom graphs. Journal of Combinatorial Theory Series B. 2026;176:254-267. doi:10.1016/j.jctb.2025.09.002","short":"M. Christoph, R. Nenadov, K.H. Petrova, Journal of Combinatorial Theory Series B 176 (2026) 254–267.","chicago":"Christoph, Micha, Rajko Nenadov, and Kalina H Petrova. “The Hamilton Space of Pseudorandom Graphs.” Journal of Combinatorial Theory Series B. Elsevier, 2026. https://doi.org/10.1016/j.jctb.2025.09.002."},"publication_identifier":{"issn":["0095-8956"],"eissn":["1096-0902"]},"doi":"10.1016/j.jctb.2025.09.002","title":"The Hamilton space of pseudorandom graphs","publication":"Journal of Combinatorial Theory Series B","quality_controlled":"1","fulldoi":"https://doi.org/10.1016/j.jctb.2025.09.002","day":"01","OA_type":"hybrid","scopus_import":"1","publication_status":"published","file_date_updated":"2026-01-05T13:29:34Z","article_type":"original","date_published":"2026-01-01T00:00:00Z","OA_place":"publisher","PlanS_conform":"1","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)"},"type":"journal_article","ec_funded":1,"month":"01","author":[{"full_name":"Christoph, Micha","last_name":"Christoph","first_name":"Micha"},{"last_name":"Nenadov","first_name":"Rajko","full_name":"Nenadov, Rajko"},{"id":"554ff4e4-f325-11ee-b0c4-a10dbd523381","last_name":"Petrova","first_name":"Kalina H","full_name":"Petrova, Kalina H"}],"date_created":"2025-10-05T22:01:34Z","has_accepted_license":"1","publisher":"Elsevier","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","acknowledgement":"This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 101034413. Image 1 Part of this research was conducted while the author was at Department of Computer Science, ETH Zürich, Switzerland. This author was supported by grant no. CRSII5 173721 of the Swiss National Science Foundation.","ddc":["510"],"oa":1,"isi":1,"project":[{"call_identifier":"H2020","_id":"fc2ed2f7-9c52-11eb-aca3-c01059dda49c","name":"IST-BRIDGE: International postdoctoral program","grant_number":"101034413"}],"corr_author":"1","abstract":[{"text":"We show that if n is odd and p>=Clog n/n, then with high probability Hamilton cycles in G(n,p) span its cycle space. More generally, we show this holds for a class of graphs satisfying certain natural pseudorandom properties. The proof is based on a novel idea of parity-switchers, which can be thought of as analogues of absorbers in the context of cycle spaces. As another application of our method, we show that Hamilton cycles in a near-Dirac graph G, that is, a graph G with odd n vertices and minimum degree n/2+C for sufficiently large constant C, span its cycle space.\r\n","lang":"eng"}],"file":[{"file_name":"2026_JourCombTheoryB_Christoph.pdf","success":1,"file_size":688924,"date_updated":"2026-01-05T13:29:34Z","date_created":"2026-01-05T13:29:34Z","creator":"dernst","relation":"main_file","access_level":"open_access","checksum":"60676af4af4b3243ba187e7d65440d99","content_type":"application/pdf","file_id":"20953"}],"oa_version":"Published Version","language":[{"iso":"eng"}],"external_id":{"arxiv":["2402.01447"],"isi":["001585783400001"]},"volume":176,"year":"2026","department":[{"_id":"MaKw"}],"intvolume":" 176","_id":"20422","page":"254-267","date_updated":"2026-01-05T13:29:52Z","status":"public","article_processing_charge":"Yes (via OA deal)"},{"type":"journal_article","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)"},"ec_funded":1,"author":[{"last_name":"Boyadzhiyska","first_name":"Simona","full_name":"Boyadzhiyska, Simona"},{"last_name":"Das","first_name":"Shagnik","full_name":"Das, Shagnik"},{"full_name":"Lesgourgues, Thomas","last_name":"Lesgourgues","first_name":"Thomas"},{"first_name":"Kalina H","id":"554ff4e4-f325-11ee-b0c4-a10dbd523381","last_name":"Petrova","full_name":"Petrova, Kalina H"}],"article_number":"104235","month":"01","date_created":"2025-10-16T13:14:34Z","OA_type":"hybrid","day":"01","fulldoi":"https://doi.org/10.1016/j.ejc.2025.104235","date_published":"2026-01-01T00:00:00Z","file_date_updated":"2026-01-05T13:34:40Z","article_type":"original","publication_status":"published","scopus_import":"1","PlanS_conform":"1","OA_place":"publisher","quality_controlled":"1","publication_identifier":{"issn":["0195-6698"]},"citation":{"short":"S. Boyadzhiyska, S. Das, T. Lesgourgues, K.H. Petrova, European Journal of Combinatorics 131 (2026).","chicago":"Boyadzhiyska, Simona, Shagnik Das, Thomas Lesgourgues, and Kalina H Petrova. “Odd-Ramsey Numbers of Complete Bipartite Graphs.” European Journal of Combinatorics. Elsevier, 2026. https://doi.org/10.1016/j.ejc.2025.104235.","ieee":"S. Boyadzhiyska, S. Das, T. Lesgourgues, and K. H. Petrova, “Odd-Ramsey numbers of complete bipartite graphs,” European Journal of Combinatorics, vol. 131. Elsevier, 2026.","mla":"Boyadzhiyska, Simona, et al. “Odd-Ramsey Numbers of Complete Bipartite Graphs.” European Journal of Combinatorics, vol. 131, 104235, Elsevier, 2026, doi:10.1016/j.ejc.2025.104235.","apa":"Boyadzhiyska, S., Das, S., Lesgourgues, T., & Petrova, K. H. (2026). Odd-Ramsey numbers of complete bipartite graphs. European Journal of Combinatorics. Elsevier. https://doi.org/10.1016/j.ejc.2025.104235","ista":"Boyadzhiyska S, Das S, Lesgourgues T, Petrova KH. 2026. Odd-Ramsey numbers of complete bipartite graphs. European Journal of Combinatorics. 131, 104235.","ama":"Boyadzhiyska S, Das S, Lesgourgues T, Petrova KH. Odd-Ramsey numbers of complete bipartite graphs. European Journal of Combinatorics. 2026;131. doi:10.1016/j.ejc.2025.104235"},"arxiv":1,"doi":"10.1016/j.ejc.2025.104235","title":"Odd-Ramsey numbers of complete bipartite graphs","publication":"European Journal of Combinatorics","department":[{"_id":"MaKw"}],"intvolume":" 131","_id":"20482","article_processing_charge":"Yes (via OA deal)","date_updated":"2026-01-05T13:34:48Z","status":"public","isi":1,"project":[{"call_identifier":"H2020","name":"IST-BRIDGE: International postdoctoral program","grant_number":"101034413","_id":"fc2ed2f7-9c52-11eb-aca3-c01059dda49c"}],"file":[{"success":1,"file_name":"2026_EuropJourCombinatorics_Boyadzhiyska.pdf","file_size":563029,"date_created":"2026-01-05T13:34:40Z","date_updated":"2026-01-05T13:34:40Z","creator":"dernst","relation":"main_file","access_level":"open_access","checksum":"52883daa217398396cbf9b8ad9ddae92","content_type":"application/pdf","file_id":"20954"}],"abstract":[{"lang":"eng","text":"In his study of graph codes, Alon introduced the concept of the odd-Ramsey number of a family of graphs H in Kn, defined as the minimum number of colours needed to colour the edges of K so that every copy of a graph H E H intersects some colour class in an odd number of edges. In this paper, we focus on complete bipartite graphs. First, we completely resolve the problem when H is the family of all spanning complete bipartite graphs on n vertices. We then focus on its subfamilies, that is, {Kt,n-t : t E T} for a fixed set of integers T c [[n/2]]. We prove that the odd-Ramsey problem is equivalent to determining the maximum dimension of a linear binary code avoiding codewords of given weights, and leverage known results from coding theory to deduce asymptotically tight bounds in our setting. We conclude with bounds for the odd-Ramsey numbers of fixed (that is, non-spanning) complete bipartite subgraphs."}],"corr_author":"1","language":[{"iso":"eng"}],"oa_version":"Published Version","volume":131,"year":"2026","external_id":{"isi":["001573380700001"],"arxiv":["2410.05887"]},"acknowledgement":"The authors would like to thank Gilles Zémor for a helpful clarification on [3], Deepak Bal and Patrick Bennett for bringing [25] to their attention, and both referees for several helpful comments.\r\nS.B.: Most of this research was conducted while the author was at the School of Mathematics, University of Birmingham, Birmingham, United Kingdom. The research leading to these results was supported by EPSRC, United Kingdom, grant no. EP/V048287/1 and by ERC Advanced Grants “GeoScape”, no. 882971 and “ERMiD”, no. 101054936. There are no additional data beyond that contained within the main manuscript.\r\nS.D.: Research supported by Taiwan NSTC grants 111-2115-M-002-009-MY2 and 113-2628-M-002-008-MY4.\r\nK.P.: This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 101034413. Parts of this research was conducted while K.P. was at the Department of Computer Science, ETH Zürich, Switzerland, supported by Swiss National Science Foundation, Switzerland , grant no. CRSII5 173721.","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","oa":1,"ddc":["500"],"publisher":"Elsevier","has_accepted_license":"1"},{"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","acknowledgement":"Z.H. was supported by SNSF grant 200021-228014. M.K. was supported by ERC Starting Grant “RANDSTRUCT” No. 101076777. L.S. was supported by the Deutsche Forschungsgemeinschaft (DFG, German\r\nResearch Foundation)—CRC 1720–539309657. This research was conducted during the period M.S. served\r\nas a Clay Research Fellow. This work began when the authors were visiting Mathematisches Forschungsinstitut Oberwolfach, which\r\nprovided ideal working conditions. M.S. thanks Vishesh Jain for initial discussions regarding the problem.\r\nWe also thank the anonymous referee for helpful comments.","oa":1,"ddc":["500"],"publisher":"Oxford University Press","has_accepted_license":"1","supplementarymaterial":"no","_id":"22147","date_updated":"2026-06-29T09:19:14Z","status":"public","article_processing_charge":"Yes (via OA deal)","department":[{"_id":"MaKw"}],"intvolume":" 2026","oa_version":"Published Version","language":[{"iso":"eng"}],"external_id":{"arxiv":["2510.12933"]},"volume":2026,"researchdata_availability":"no","year":"2026","project":[{"grant_number":"101076777","name":"Randomness and structure in combinatorics","_id":"bd95085b-d553-11ed-ba76-e55d3349be45"}],"corr_author":"1","abstract":[{"text":"Let 1 ≤ k ≤ n and M be a random n × n matrix with independent uniformly random {±1}-entries. We\r\nshow that there exists an absolute constant c > 0 such that\r\nP[rank(M) ≤ n − k] ≤ exp(−cnk).\r\nThis confirms a well-known prediction in the area, extending a result of Rudelson (who previously\r\nproved this same result under the restriction k ≤ √n, via different methods).","lang":"eng"}],"file":[{"date_created":"2026-06-29T09:15:15Z","date_updated":"2026-06-29T09:15:15Z","creator":"dernst","file_size":524993,"success":1,"file_name":"2026_IMRN_Hunter.pdf","checksum":"396b47d0532d7ea509f8cd30f11392a8","content_type":"application/pdf","file_id":"22151","relation":"main_file","access_level":"open_access"}],"issue":"12","quality_controlled":"1","title":"On random matrices with large corank","publication":"International Mathematics Research Notices","arxiv":1,"publication_identifier":{"issn":["1073-7928"],"eissn":["1687-0247"]},"citation":{"ieee":"Z. Hunter, M. A. Kwan, L. Sauermann, and M. Sawhney, “On random matrices with large corank,” International Mathematics Research Notices, vol. 2026, no. 12. Oxford University Press, 2026.","mla":"Hunter, Zach, et al. “On Random Matrices with Large Corank.” International Mathematics Research Notices, vol. 2026, no. 12, rnag126, Oxford University Press, 2026, doi:10.1093/imrn/rnag126.","ista":"Hunter Z, Kwan MA, Sauermann L, Sawhney M. 2026. On random matrices with large corank. International Mathematics Research Notices. 2026(12), rnag126.","apa":"Hunter, Z., Kwan, M. A., Sauermann, L., & Sawhney, M. (2026). On random matrices with large corank. International Mathematics Research Notices. Oxford University Press. https://doi.org/10.1093/imrn/rnag126","ama":"Hunter Z, Kwan MA, Sauermann L, Sawhney M. On random matrices with large corank. International Mathematics Research Notices. 2026;2026(12). doi:10.1093/imrn/rnag126","short":"Z. Hunter, M.A. Kwan, L. Sauermann, M. Sawhney, International Mathematics Research Notices 2026 (2026).","chicago":"Hunter, Zach, Matthew Alan Kwan, Lisa Sauermann, and Mehtaab Sawhney. “On Random Matrices with Large Corank.” International Mathematics Research Notices. Oxford University Press, 2026. https://doi.org/10.1093/imrn/rnag126."},"doi":"10.1093/imrn/rnag126","month":"06","article_number":"rnag126","author":[{"full_name":"Hunter, Zach","first_name":"Zach","last_name":"Hunter"},{"full_name":"Kwan, Matthew Alan","first_name":"Matthew Alan","orcid":"0000-0002-4003-7567","last_name":"Kwan","id":"5fca0887-a1db-11eb-95d1-ca9d5e0453b3"},{"full_name":"Sauermann, Lisa","last_name":"Sauermann","first_name":"Lisa"},{"first_name":"Mehtaab","last_name":"Sawhney","full_name":"Sawhney, Mehtaab"}],"date_created":"2026-06-28T22:01:35Z","type":"journal_article","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)"},"das_tickbox":"0","OA_place":"publisher","PlanS_conform":"1","fulldoi":"https://doi.org/10.1093/imrn/rnag126","day":"01","OA_type":"hybrid","scopus_import":"1","publication_status":"published","article_type":"original","file_date_updated":"2026-06-29T09:15:15Z","date_published":"2026-06-01T00:00:00Z"},{"doi":"10.1112/blms.70464","arxiv":1,"publication_identifier":{"eissn":["1469-2120"],"issn":["0024-6093"]},"citation":{"ieee":"M. Anastos, J. Erde, M. Kang, and V. Pfenninger, “A short proof of a central limit theorem for the order of the giant component and k-core,” Bulletin of the London Mathematical Society, vol. 58, no. 8. Wiley, 2026.","ama":"Anastos M, Erde J, Kang M, Pfenninger V. A short proof of a central limit theorem for the order of the giant component and k-core. Bulletin of the London Mathematical Society. 2026;58(8). doi:10.1112/blms.70464","apa":"Anastos, M., Erde, J., Kang, M., & Pfenninger, V. (2026). A short proof of a central limit theorem for the order of the giant component and k-core. Bulletin of the London Mathematical Society. Wiley. https://doi.org/10.1112/blms.70464","ista":"Anastos M, Erde J, Kang M, Pfenninger V. 2026. A short proof of a central limit theorem for the order of the giant component and k-core. Bulletin of the London Mathematical Society. 58(8), e70464.","mla":"Anastos, Michael, et al. “A Short Proof of a Central Limit Theorem for the Order of the Giant Component and K-Core.” Bulletin of the London Mathematical Society, vol. 58, no. 8, e70464, Wiley, 2026, doi:10.1112/blms.70464.","short":"M. Anastos, J. Erde, M. Kang, V. Pfenninger, Bulletin of the London Mathematical Society 58 (2026).","chicago":"Anastos, Michael, Joshua Erde, Mihyun Kang, and Vincent Pfenninger. “A Short Proof of a Central Limit Theorem for the Order of the Giant Component and K-Core.” Bulletin of the London Mathematical Society. Wiley, 2026. https://doi.org/10.1112/blms.70464."},"publication":"Bulletin of the London Mathematical Society","title":"A short proof of a central limit theorem for the order of the giant component and k-core","quality_controlled":"1","issue":"8","scopus_import":"1","article_type":"original","file_date_updated":"2026-09-07T13:53:26Z","date_published":"2026-08-01T00:00:00Z","publication_status":"published","fulldoi":"https://doi.org/10.1112/blms.70464","OA_type":"hybrid","day":"01","OA_place":"publisher","PlanS_conform":"1","das_tickbox":"0","type":"journal_article","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)"},"date_created":"2026-08-23T22:01:47Z","month":"08","article_number":"e70464","author":[{"first_name":"Michael","last_name":"Anastos","id":"0b2a4358-bb35-11ec-b7b9-e3279b593dbb","full_name":"Anastos, Michael"},{"last_name":"Erde","first_name":"Joshua","full_name":"Erde, Joshua"},{"full_name":"Kang, Mihyun","first_name":"Mihyun","last_name":"Kang"},{"first_name":"Vincent","last_name":"Pfenninger","full_name":"Pfenninger, Vincent"}],"mathsc":["05C80","60F05"],"supplementarymaterial":"no","has_accepted_license":"1","publisher":"Wiley","ddc":["510"],"oa":1,"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","acknowledgement":"This research was funded in whole or in part by the Austrian Science Fund (FWF) [10.55776/P36131 (Joshua Erde), 10.55776/I6502 (Mihyun Kang), 10.55776/ESP3863424 (Michael Anastos)] and by the Swiss National Science Foundation (SNSF) [P500-2_235474] (Vincent Pfenninger). For open access purposes, the authors have applied a CC BY public copyright license to any author accepted manuscript version arising from this submission. The authors thank the reviewers for helpful comments on the paper, and for bringing the particular form of Theorem 2.2 to our attention.","abstract":[{"lang":"eng","text":"In this note we outline a new and simple approach to proving central limit theorems for various ‘global’ graph parameters that have robust ‘local’ approximations, using the Efron–Stein inequality, which relies on a combinatorial analysis of the stability of these approximations under resampling an edge. As an application, we give short proofs of a central limit theorem for the order of the giant component and of the 𝑘\r\n-core for sparse random graphs."}],"file":[{"file_name":"2026_BulletinLondonMathSoc_Anastos.pdf","file_size":248641,"success":1,"creator":"dernst","date_created":"2026-09-07T13:53:26Z","date_updated":"2026-09-07T13:53:26Z","access_level":"open_access","relation":"main_file","content_type":"application/pdf","file_id":"22845","checksum":"b4b21f05c3fc62fde243a5ddd46cef49"}],"project":[{"_id":"8f906bd2-16d5-11f0-9cad-e07be8aa9ac9","grant_number":"ESP3863424","name":"Combinatorial Optimisation Problems on Sparse Random Graphs"}],"external_id":{"arxiv":["2506.11651"]},"researchdata_availability":"no","year":"2026","volume":58,"oa_version":"Published Version","language":[{"iso":"eng"}],"intvolume":" 58","department":[{"_id":"MaKw"}],"status":"public","date_updated":"2026-09-07T13:56:17Z","article_processing_charge":"Yes (in subscription journal)","_id":"22752"},{"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","acknowledgement":"Open access funding provided by Institute of Science and Technology (IST Austria). Vishesh Jain was partially supported by NSF CAREER award DMS-2237646. Matthew Kwan was supported by ERC Starting Grant “RANDSTRUCT” No. 101076777. Marcus Michelen was supported in part by NSF grants DMS-2137623 and DMS-2246624.","oa":1,"ddc":["500"],"publisher":"Springer","supplementarymaterial":"no","has_accepted_license":"1","department":[{"_id":"MaKw"}],"intvolume":" 407","_id":"22813","date_updated":"2026-09-09T12:35:24Z","status":"public","article_processing_charge":"Yes (via OA deal)","project":[{"name":"Randomness and structure in combinatorics","grant_number":"101076777","_id":"bd95085b-d553-11ed-ba76-e55d3349be45"}],"corr_author":"1","abstract":[{"lang":"eng","text":"Consider a bipartite quantum system, where Alice and Bob jointly possess\r\na pure state |ψ\u0002. Using local quantum operations on their respective subsystems, and\r\nunlimited classical communication, Alice and Bob may be able to transform |ψ\u0002 into\r\nanother state |φ\u0002. Famously, Nielsen’s theorem [28] provides a necessary and sufficient\r\nalgebraic criterion for such a transformation to be possible (namely, the entanglement\r\nspectrum of |φ\u0002 should majorise the entanglement spectrum of |ψ\u0002). In the paper where\r\nNielsen proved this theorem, he conjectured that in the limit of large dimensionality, for\r\nalmost all pairs of states |ψ\u0002, |φ\u0002 (according to the natural unitary invariant measure) such\r\na transformation is not possible. That is to say, typical pairs of quantum states |ψ\u0002, |φ\u0002\r\nare entangled in fundamentally different ways, that cannot be converted to each other via\r\nlocal operations and classical communication.ViaNielsen’s theorem, this conjecture can\r\nbe equivalently stated as a conjecture about majorisation of spectra of random matrices\r\nfrom the so-called trace-normalised complex Wishart–Laguerre ensemble. Concretely,\r\nlet X and Y be independent n × m random matrices whose entries are i.i.d. standard\r\ncomplex Gaussians; then Nielsen’s conjecture says that the probability that the spectrum\r\nof XX†/ tr(XX†) majorises the spectrum of YY†/ tr(YY†) tends to zero as both n and\r\nm grow large. We prove this conjecture, and we also confirm some related predictions\r\nof Cunden et al. [12]."}],"dataavailabilitystatement":"No datasets were used in this research.","file":[{"relation":"main_file","access_level":"open_access","checksum":"61c226b6314a88db25e864e4218d5de3","content_type":"application/pdf","file_id":"22888","file_name":"2026_CommMathPhysics_Jain.pdf","success":1,"file_size":463477,"date_updated":"2026-09-09T12:32:53Z","date_created":"2026-09-09T12:32:53Z","creator":"dernst"}],"oa_version":"Published Version","language":[{"iso":"eng"}],"external_id":{"arxiv":["2406.03335"],"pmid":["42668663"]},"year":"2026","researchdata_availability":"no","volume":407,"issue":"10","quality_controlled":"1","pmid":1,"arxiv":1,"citation":{"ama":"Jain V, Kwan MA, Michelen M. Entangled states are typically incomparable. Communications in Mathematical Physics. 2026;407(10). doi:10.1007/s00220-026-05711-4","ista":"Jain V, Kwan MA, Michelen M. 2026. Entangled states are typically incomparable. Communications in Mathematical Physics. 407(10), 214.","apa":"Jain, V., Kwan, M. A., & Michelen, M. (2026). Entangled states are typically incomparable. Communications in Mathematical Physics. Springer. https://doi.org/10.1007/s00220-026-05711-4","mla":"Jain, Vishesh, et al. “Entangled States Are Typically Incomparable.” Communications in Mathematical Physics, vol. 407, no. 10, 214, Springer, 2026, doi:10.1007/s00220-026-05711-4.","ieee":"V. Jain, M. A. Kwan, and M. Michelen, “Entangled states are typically incomparable,” Communications in Mathematical Physics, vol. 407, no. 10. Springer, 2026.","chicago":"Jain, Vishesh, Matthew Alan Kwan, and Marcus Michelen. “Entangled States Are Typically Incomparable.” Communications in Mathematical Physics. Springer, 2026. https://doi.org/10.1007/s00220-026-05711-4.","short":"V. Jain, M.A. Kwan, M. Michelen, Communications in Mathematical Physics 407 (2026)."},"publication_identifier":{"eissn":["1432-0916"],"issn":["0010-3616"]},"doi":"10.1007/s00220-026-05711-4","title":"Entangled states are typically incomparable","publication":"Communications in Mathematical Physics","type":"journal_article","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)"},"month":"08","article_number":"214","author":[{"last_name":"Jain","first_name":"Vishesh","full_name":"Jain, Vishesh"},{"orcid":"0000-0002-4003-7567","first_name":"Matthew Alan","id":"5fca0887-a1db-11eb-95d1-ca9d5e0453b3","last_name":"Kwan","full_name":"Kwan, Matthew Alan"},{"last_name":"Michelen","first_name":"Marcus","full_name":"Michelen, Marcus"}],"date_created":"2026-09-06T22:01:56Z","fulldoi":"https://doi.org/10.1007/s00220-026-05711-4","day":"28","OA_type":"hybrid","scopus_import":"1","file_date_updated":"2026-09-09T12:32:53Z","date_published":"2026-08-28T00:00:00Z","publication_status":"published","article_type":"original","das_tickbox":"1","OA_place":"publisher","PlanS_conform":"1"},{"abstract":[{"lang":"eng","text":"A famous conjecture by Itai and Zehavi states that, for every 𝑑\r\n-vertex-connected graph 𝐺\r\n and every vertex 𝑟\r\n in 𝐺\r\n, there are 𝑑\r\n spanning trees of 𝐺\r\n such that, for every vertex 𝑣\r\n in 𝐺 ∖{𝑟}\r\n, the paths between 𝑟\r\n and 𝑣\r\n in different trees are internally vertex-disjoint. We show that with high probability the Itai–Zehavi conjecture holds asymptotically for the Erdős–Rényi random graph 𝐺(𝑛,𝑝)\r\n when 𝑛𝑝 =𝜔(log𝑛)\r\n and for random regular graphs 𝐺(𝑛,𝑑)\r\n when 𝑑 =𝜔(log𝑛)\r\n. Moreover, we essentially confirm the conjecture up to a constant factor for sparser random regular graphs. This answers a question of Draganić and Krivelevich positively. Our proof makes use of recent developments on sprinkling techniques in random regular graphs."}],"file":[{"success":1,"file_size":457581,"file_name":"2026_RandomStructAlgorithms_Hollom.pdf","creator":"dernst","date_updated":"2026-10-07T07:10:43Z","date_created":"2026-10-07T07:10:43Z","access_level":"open_access","relation":"main_file","content_type":"application/pdf","file_id":"23068","checksum":"9a786460a47542f24576e947987e558a"}],"project":[{"_id":"bd95085b-d553-11ed-ba76-e55d3349be45","grant_number":"101076777","name":"Randomness and structure in combinatorics"}],"external_id":{"arxiv":["2506.23970"]},"year":"2026","volume":69,"researchdata_availability":"no","oa_version":"Published Version","language":[{"iso":"eng"}],"intvolume":" 69","department":[{"_id":"MaKw"},{"_id":"GradSch"}],"status":"public","date_updated":"2026-10-07T07:13:30Z","article_processing_charge":"Yes (via OA deal)","_id":"23001","supplementarymaterial":"yes","publisher":"Wiley","has_accepted_license":"1","ddc":["500"],"oa":1,"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","acknowledgement":"Hollom was supported by the Internal Graduate Studentship of Trinity College, Cambridge. Mond was supported by UK Research and Innovation grant MR/W007320/2. Wang was supported by the ERC Starting Grant “RANDSTRUCT” No. 101076777. Lichev was supported by the Austrian Science Fund (FWF) [10.55776/ESP624]. For open access purposes, the authors have applied a CC BY public copyright license to any author-accepted manuscript version arising from this submission. Open Access funding provided by Technische Universitat Wien.\r\n\r\nPart of this research was done during a visit of the fourth author to IST Austria. We thank IST Austria for its hospitality. We also thank the anonymous referees for their comments and suggestions. Open Access funding provided by Technische Universitat Wien. This work was supported by the European Research Council, UK Research and Innovation, the Austrian Science Fund, and Trinity College.","scopus_import":"1","file_date_updated":"2026-10-07T07:10:43Z","date_published":"2026-09-01T00:00:00Z","publication_status":"published","article_type":"original","fulldoi":"https://doi.org/10.1002/rsa.70095","day":"01","OA_type":"hybrid","OA_place":"publisher","PlanS_conform":"1","das_tickbox":"0","type":"journal_article","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)"},"date_created":"2026-09-27T22:01:52Z","month":"09","article_number":"e70095","author":[{"full_name":"Hollom, Lawrence","first_name":"Lawrence","last_name":"Hollom"},{"full_name":"Lichev, Lyuben","id":"9aa8388e-d003-11ee-8458-c4c1d7447977","last_name":"Lichev","first_name":"Lyuben"},{"last_name":"Mond","first_name":"Adva","full_name":"Mond, Adva"},{"full_name":"Portier, Julien","last_name":"Portier","first_name":"Julien"},{"id":"1917d194-076e-11ed-97cd-837255f88785","last_name":"Wang","first_name":"Yiting","full_name":"Wang, Yiting"}],"doi":"10.1002/rsa.70095","arxiv":1,"citation":{"chicago":"Hollom, Lawrence, Lyuben Lichev, Adva Mond, Julien Portier, and Yiting Wang. “Approximate Itai–Zehavi Conjecture for Random Graphs.” Random Structures and Algorithms. Wiley, 2026. https://doi.org/10.1002/rsa.70095.","short":"L. Hollom, L. Lichev, A. Mond, J. Portier, Y. Wang, Random Structures and Algorithms 69 (2026).","mla":"Hollom, Lawrence, et al. “Approximate Itai–Zehavi Conjecture for Random Graphs.” Random Structures and Algorithms, vol. 69, no. 2, e70095, Wiley, 2026, doi:10.1002/rsa.70095.","ista":"Hollom L, Lichev L, Mond A, Portier J, Wang Y. 2026. Approximate Itai–Zehavi conjecture for random graphs. Random Structures and Algorithms. 69(2), e70095.","ama":"Hollom L, Lichev L, Mond A, Portier J, Wang Y. Approximate Itai–Zehavi conjecture for random graphs. Random Structures and Algorithms. 2026;69(2). doi:10.1002/rsa.70095","apa":"Hollom, L., Lichev, L., Mond, A., Portier, J., & Wang, Y. (2026). Approximate Itai–Zehavi conjecture for random graphs. Random Structures and Algorithms. Wiley. https://doi.org/10.1002/rsa.70095","ieee":"L. Hollom, L. Lichev, A. Mond, J. Portier, and Y. Wang, “Approximate Itai–Zehavi conjecture for random graphs,” Random Structures and Algorithms, vol. 69, no. 2. Wiley, 2026."},"publication_identifier":{"issn":["1042-9832"],"eissn":["1098-2418"]},"publication":"Random Structures and Algorithms","title":"Approximate Itai–Zehavi conjecture for random graphs","quality_controlled":"1","issue":"2"},{"publication":"Bulletin of the London Mathematical Society","title":"No‐(k+1)‐in‐line problem for large constant k","doi":"10.1112/blms.70510","publication_identifier":{"issn":["0024-6093"],"eissn":["1469-2120"]},"citation":{"short":"A. Grebennikov, M.A. Kwan, Bulletin of the London Mathematical Society 58 (2026).","chicago":"Grebennikov, Aleksandr, and Matthew Alan Kwan. “No‐(K+1)‐in‐line Problem for Large Constant K.” Bulletin of the London Mathematical Society. Wiley, 2026. https://doi.org/10.1112/blms.70510.","ieee":"A. Grebennikov and M. A. Kwan, “No‐(k+1)‐in‐line problem for large constant k,” Bulletin of the London Mathematical Society, vol. 58, no. 10. Wiley, 2026.","ista":"Grebennikov A, Kwan MA. 2026. No‐(k+1)‐in‐line problem for large constant k. Bulletin of the London Mathematical Society. 58(10), e70510.","ama":"Grebennikov A, Kwan MA. No‐(k+1)‐in‐line problem for large constant k. Bulletin of the London Mathematical Society. 2026;58(10). doi:10.1112/blms.70510","apa":"Grebennikov, A., & Kwan, M. A. (2026). No‐(k+1)‐in‐line problem for large constant k. Bulletin of the London Mathematical Society. Wiley. https://doi.org/10.1112/blms.70510","mla":"Grebennikov, Aleksandr, and Matthew Alan Kwan. “No‐(K+1)‐in‐line Problem for Large Constant K.” Bulletin of the London Mathematical Society, vol. 58, no. 10, e70510, Wiley, 2026, doi:10.1112/blms.70510."},"quality_controlled":"1","issue":"10","OA_place":"publisher","PlanS_conform":"1","das_tickbox":"0","scopus_import":"1","date_published":"2026-10-01T00:00:00Z","article_type":"original","publication_status":"published","file_date_updated":"2026-10-14T08:37:16Z","fulldoi":"https://doi.org/10.1112/blms.70510","OA_type":"hybrid","day":"01","date_created":"2026-10-12T13:57:47Z","month":"10","article_number":"e70510","author":[{"full_name":"Grebennikov, Aleksandr","first_name":"Aleksandr","id":"776468ec-50b0-11ef-861e-96ad19e894f3","last_name":"Grebennikov"},{"orcid":"0000-0002-4003-7567","first_name":"Matthew Alan","id":"5fca0887-a1db-11eb-95d1-ca9d5e0453b3","last_name":"Kwan","full_name":"Kwan, Matthew Alan"}],"type":"journal_article","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)"},"has_accepted_license":"1","supplementarymaterial":"no","publisher":"Wiley","mathsc":["05D40","52C10","52C35"],"ddc":["510"],"oa":1,"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","acknowledgement":"We would like to thank Huy Pham for insightful discussions about the proof approach, and wewould like to thank the anonymous referees for their careful reading and helpful comments. Bothauthors are supported by ERC Starting Grant “RANDSTRUCT” No. 101076777.Open Access funding provided by Institute of Science and Technology Austria.","volume":58,"year":"2026","researchdata_availability":"no","oa_version":"Published Version","language":[{"iso":"eng"}],"corr_author":"1","abstract":[{"lang":"eng","text":"How many points can be placed in an 𝑛 × 𝑛 grid sothat every (affine) line contains at most 𝑘 points? Weprove that for 𝑛 ⩾ 𝑘 ⩾ 1037 the maximum number ofpoints is exactly 𝑘𝑛. Our proof builds on the recent workof Kovács, Nagy and Szabó (who proved an analogousresult when 𝑘 is at least about √𝑛 log 𝑛), incorporat-ing ideas of Jain and Pham. Using the same approach,we also obtain new bounds for higher-dimensionalextensions of this problem"}],"file":[{"success":1,"file_name":"2026_BulletinLondonMathSoc_Grebennikov.pdf","file_size":284851,"creator":"dernst","date_created":"2026-10-14T08:37:16Z","date_updated":"2026-10-14T08:37:16Z","access_level":"open_access","relation":"main_file","content_type":"application/pdf","file_id":"23139","checksum":"cce830e3fc66174803b91103f5b88a29"}],"project":[{"_id":"bd95085b-d553-11ed-ba76-e55d3349be45","name":"Randomness and structure in combinatorics","grant_number":"101076777"}],"date_updated":"2026-10-14T08:41:00Z","status":"public","article_processing_charge":"Yes (via OA deal)","_id":"23123","intvolume":" 58","department":[{"_id":"MaKw"},{"_id":"GradSch"}]},{"publication":"Annals of Applied Probability","title":"Monotonicity and decompositions of random regular graphs","doi":"10.1214/26-AAP2345","publication_identifier":{"eissn":["2168-8737"],"issn":["1050-5164"]},"citation":{"short":"L. Hollom, L. Lichev, A. Mond, J. Portier, Y. Wang, Annals of Applied Probability 36 (2026) 4716–4738.","chicago":"Hollom, Lawrence, Lyuben Lichev, Adva Mond, Julien Portier, and Yiting Wang. “Monotonicity and Decompositions of Random Regular Graphs.” Annals of Applied Probability. Institute of Mathematical Statistics, 2026. https://doi.org/10.1214/26-AAP2345.","ieee":"L. Hollom, L. Lichev, A. Mond, J. Portier, and Y. Wang, “Monotonicity and decompositions of random regular graphs,” Annals of Applied Probability, vol. 36, no. 5. Institute of Mathematical Statistics, pp. 4716–4738, 2026.","mla":"Hollom, Lawrence, et al. “Monotonicity and Decompositions of Random Regular Graphs.” Annals of Applied Probability, vol. 36, no. 5, Institute of Mathematical Statistics, 2026, pp. 4716–38, doi:10.1214/26-AAP2345.","ista":"Hollom L, Lichev L, Mond A, Portier J, Wang Y. 2026. Monotonicity and decompositions of random regular graphs. Annals of Applied Probability. 36(5), 4716–4738.","ama":"Hollom L, Lichev L, Mond A, Portier J, Wang Y. Monotonicity and decompositions of random regular graphs. Annals of Applied Probability. 2026;36(5):4716-4738. doi:10.1214/26-AAP2345","apa":"Hollom, L., Lichev, L., Mond, A., Portier, J., & Wang, Y. (2026). Monotonicity and decompositions of random regular graphs. Annals of Applied Probability. Institute of Mathematical Statistics. https://doi.org/10.1214/26-AAP2345"},"arxiv":1,"quality_controlled":"1","issue":"5","PlanS_conform":"1","OA_place":"publisher","das_tickbox":"0","publication_status":"published","article_type":"original","file_date_updated":"2026-10-19T07:33:58Z","date_published":"2026-10-01T00:00:00Z","scopus_import":"1","OA_type":"hybrid","day":"01","fulldoi":"https://doi.org/10.1214/26-AAP2345","date_created":"2026-10-18T22:01:52Z","author":[{"full_name":"Hollom, Lawrence","last_name":"Hollom","first_name":"Lawrence"},{"id":"9aa8388e-d003-11ee-8458-c4c1d7447977","last_name":"Lichev","first_name":"Lyuben","full_name":"Lichev, Lyuben"},{"first_name":"Adva","last_name":"Mond","full_name":"Mond, Adva"},{"full_name":"Portier, Julien","last_name":"Portier","first_name":"Julien"},{"first_name":"Yiting","last_name":"Wang","id":"1917d194-076e-11ed-97cd-837255f88785","full_name":"Wang, Yiting"}],"month":"10","type":"journal_article","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)"},"supplementarymaterial":"no","has_accepted_license":"1","publisher":"Institute of Mathematical Statistics","mathsc":["05C80"],"oa":1,"ddc":["500"],"acknowledgement":"We are grateful to Béla Bollobás, to Matthew Kwan and to Fabio\r\nToninelli for their helpful comments and suggestions, and to Nicholas Wormald for useful\r\nremarks on a preliminary version of this work and pointing us towards the reference [29].\r\nThe authors would also like to thank several anonymous referees for many useful suggestions,\r\nincluding pointing out that Lemma 2.10 already appeared in [11].\r\nPart of this research was done during a visit of the fourth author to IST Austria. We thank\r\nIST Austria for its hospitality.\r\nFor open access purposes, the authors have applied a CC BY public copyright license to\r\nany author accepted manuscript version arising from this submission. L. Hollom was supported by the Internal Graduate Studentship of Trinity College, Cambridge.\r\nL. Lichev was supported by the Austrian Science Fund (FWF) grant No. 10.55776/ESP624.\r\nA. Mond was supported by UK Research and Innovation grant MR/W007320/2. Wang was\r\nsupported by the ERC Starting Grant “RANDSTRUCT” No. 101076777.","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","volume":36,"researchdata_availability":"no","year":"2026","external_id":{"arxiv":["2505.22875"]},"keyword":["Random regular graphs","coupling","contiguity","total variation distance","decomposition"],"language":[{"iso":"eng"}],"oa_version":"Published Version","file":[{"creator":"dernst","date_created":"2026-10-19T07:33:58Z","date_updated":"2026-10-19T07:33:58Z","file_size":320593,"success":1,"file_name":"2026_AnnalsAppliedProbability_Hollom.pdf","content_type":"application/pdf","file_id":"23165","checksum":"46101e61d29cf40f0a204801d15a848f","access_level":"open_access","relation":"main_file"}],"abstract":[{"lang":"eng","text":"In this work we establish several monotonicity and decomposition results in the framework of random regular graphs. Among other results, we\r\nshow that, for a wide range of parameters d1 ≤ d2, there exists a coupling of\r\nG(n,d1) and G(n,d2) satisfying that G(n,d1) ⊆ G(n,d2) with high probability, confirming a conjecture of Gao, Isaev and McKay in a new regime.\r\nOur contributions include new tools for analysing contiguity and total variation distance between random regular graph models, a novel procedure for\r\ngenerating unions of random edge-disjoint perfect matchings, and refined estimates of Gao’s bounds on the number of perfect matchings in random regular graphs. In addition, we make progress towards another conjecture of Isaev,\r\nMcKay, Southwell and Zhukovskii."}],"project":[{"_id":"bd95085b-d553-11ed-ba76-e55d3349be45","grant_number":"101076777","name":"Randomness and structure in combinatorics"},{"name":"THRESHOLDS IN SEMI-RANDOM PROCESSES AND GRAPH EMBEDDINGS","grant_number":"E624","_id":"e19fef4c-1f3d-11f0-ac04-b8b1af6886d8"}],"article_processing_charge":"Yes (in subscription journal)","status":"public","date_updated":"2026-10-19T07:39:17Z","page":"4716-4738","_id":"23155","intvolume":" 36","department":[{"_id":"MaKw"},{"_id":"GradSch"}]},{"title":"A note on high-dimensional discrepancy of subtrees","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2412.04170"}],"publication":"Discrete and Computational Geometry","arxiv":1,"publication_identifier":{"issn":["0179-5376"],"eissn":["1432-0444"]},"citation":{"ieee":"L. Hollom, L. Lichev, A. Mond, and J. Portier, “A note on high-dimensional discrepancy of subtrees,” Discrete and Computational Geometry. Springer, 2026.","mla":"Hollom, Lawrence, et al. “A Note on High-Dimensional Discrepancy of Subtrees.” Discrete and Computational Geometry, Springer, 2026, doi:10.1007/s00454-025-00790-x.","ama":"Hollom L, Lichev L, Mond A, Portier J. A note on high-dimensional discrepancy of subtrees. Discrete and Computational Geometry. 2026. doi:10.1007/s00454-025-00790-x","ista":"Hollom L, Lichev L, Mond A, Portier J. 2026. A note on high-dimensional discrepancy of subtrees. Discrete and Computational Geometry.","apa":"Hollom, L., Lichev, L., Mond, A., & Portier, J. (2026). A note on high-dimensional discrepancy of subtrees. Discrete and Computational Geometry. Springer. https://doi.org/10.1007/s00454-025-00790-x","short":"L. Hollom, L. Lichev, A. Mond, J. Portier, Discrete and Computational Geometry (2026).","chicago":"Hollom, Lawrence, Lyuben Lichev, Adva Mond, and Julien Portier. “A Note on High-Dimensional Discrepancy of Subtrees.” Discrete and Computational Geometry. Springer, 2026. https://doi.org/10.1007/s00454-025-00790-x."},"doi":"10.1007/s00454-025-00790-x","quality_controlled":"1","das_tickbox":"1","OA_place":"repository","fulldoi":"https://doi.org/10.1007/s00454-025-00790-x","OA_type":"green","day":"09","scopus_import":"1","publication_status":"epub_ahead","date_published":"2026-10-09T00:00:00Z","article_type":"original","month":"10","author":[{"last_name":"Hollom","first_name":"Lawrence","full_name":"Hollom, Lawrence"},{"first_name":"Lyuben","last_name":"Lichev","id":"9aa8388e-d003-11ee-8458-c4c1d7447977","full_name":"Lichev, Lyuben"},{"full_name":"Mond, Adva","last_name":"Mond","first_name":"Adva"},{"first_name":"Julien","last_name":"Portier","full_name":"Portier, Julien"}],"date_created":"2026-10-18T22:01:53Z","type":"journal_article","publisher":"Springer","supplementarymaterial":"no","mathsc":["05C05","05C35","11K38"],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","acknowledgement":"The authors would like to thank both Béla Bollobás and the anonymous reviewer for their careful readings of the manuscript and for their valuable comments. The first author was funded by Trinity College, Cambridge. The second author was funded in whole or in part by the Austrian Science Fund (FWF) 10.55776/ESP624. The third author was supported by UK Research and Innovation grant MR/W007320/2. The fourth author was funded by EPSRC (Engineering and Physical Sciences Research Council) and by the Cambridge Commonwealth, European and International Trust.","oa":1,"oa_version":"Preprint","language":[{"iso":"eng"}],"keyword":["Trees","High-dimensional discrepancy","Oriented discrepancy"],"external_id":{"arxiv":["2412.04170"]},"year":"2026","researchdata_availability":"no","project":[{"_id":"e19fef4c-1f3d-11f0-ac04-b8b1af6886d8","grant_number":"E624","name":"THRESHOLDS IN SEMI-RANDOM PROCESSES AND GRAPH EMBEDDINGS"}],"abstract":[{"text":"For a tree T and a function f:E(T)→Sd, the imbalance of a subtree T′⊆T is given by |∑e∈E(T′)f(e)|. The d-dimensional discrepancy of the tree T is the minimum, over all functions f as above, of the maximum imbalance of a subtree of T. We prove tight asymptotic bounds for the discrepancy of a tree T, confirming a conjecture of Krishna, Michaeli, Sarantis, Wang and Wang. We also settle a related conjecture on oriented discrepancy of subtrees by the same authors.","lang":"eng"}],"dataavailabilitystatement":"Data sharing not applicable to this article as no datasets were generated or analysed during the current study.","_id":"23158","date_updated":"2026-10-19T08:20:42Z","status":"public","article_processing_charge":"No","department":[{"_id":"MaKw"}]},{"language":[{"iso":"eng"}],"oa_version":"Preprint","year":"2025","external_id":{"arxiv":["2508.00480"]},"project":[{"call_identifier":"H2020","grant_number":"101034413","name":"IST-BRIDGE: International postdoctoral program","_id":"fc2ed2f7-9c52-11eb-aca3-c01059dda49c"}],"abstract":[{"text":"We show that, for any graph F and η > 0, there exists a d0 = d0(F, η) such that every nvertex d-regular graph with d ≥ d0 has a collection of vertex-disjoint F-subdivisions covering\r\nat least (1 − η)n vertices. This verifies a conjecture of Verstraëte from 2002 and improves a\r\nrecent result of Letzter, Methuku and Sudakov which additionally required d to be at least\r\npolylogarithmic in n.\r\n","lang":"eng"}],"_id":"21211","article_processing_charge":"No","status":"public","date_updated":"2026-02-10T11:39:16Z","department":[{"_id":"MaKw"}],"acknowledgement":"Supported by the European Research Council (ERC) under the European Union Horizon 2020 research and innovation programme (grant agreement No. 947978). This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 101034413.","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","oa":1,"OA_place":"repository","OA_type":"green","day":"01","fulldoi":"https://doi.org/10.48550/arXiv.2508.00480","publication_status":"submitted","date_published":"2025-08-01T00:00:00Z","author":[{"last_name":"Montgomery","first_name":"Richard","full_name":"Montgomery, Richard"},{"full_name":"Petrova, Kalina H","id":"554ff4e4-f325-11ee-b0c4-a10dbd523381","last_name":"Petrova","first_name":"Kalina H"},{"full_name":"Ranganathan, Arjun","last_name":"Ranganathan","first_name":"Arjun"},{"full_name":"Tan, Jane","last_name":"Tan","first_name":"Jane"}],"month":"08","article_number":"2508.00480","date_created":"2026-02-10T11:32:41Z","type":"preprint","ec_funded":1,"title":"Packing subdivisions into regular graphs","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2508.00480"}],"publication":"arXiv","citation":{"chicago":"Montgomery, Richard, Kalina H Petrova, Arjun Ranganathan, and Jane Tan. “Packing Subdivisions into Regular Graphs.” ArXiv, n.d. https://doi.org/10.48550/arXiv.2508.00480.","short":"R. Montgomery, K.H. Petrova, A. Ranganathan, J. Tan, ArXiv (n.d.).","ista":"Montgomery R, Petrova KH, Ranganathan A, Tan J. Packing subdivisions into regular graphs. arXiv, 2508.00480.","apa":"Montgomery, R., Petrova, K. H., Ranganathan, A., & Tan, J. (n.d.). Packing subdivisions into regular graphs. arXiv. https://doi.org/10.48550/arXiv.2508.00480","ama":"Montgomery R, Petrova KH, Ranganathan A, Tan J. Packing subdivisions into regular graphs. arXiv. doi:10.48550/arXiv.2508.00480","mla":"Montgomery, Richard, et al. “Packing Subdivisions into Regular Graphs.” ArXiv, 2508.00480, doi:10.48550/arXiv.2508.00480.","ieee":"R. Montgomery, K. H. Petrova, A. Ranganathan, and J. Tan, “Packing subdivisions into regular graphs,” arXiv. ."},"arxiv":1,"doi":"10.48550/arXiv.2508.00480"},{"oa":1,"ddc":["510"],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","acknowledgement":"We would like to thank the anonymous referee for a number of helpful comments and suggestions. Matthew Kwan was supported by ERC Starting Grant “RANDSTRUCT” No. 101076777. Lisa Sauermann was supported in part by NSF Award DMS-2100157 and a Sloan Research Fellowship, and in part by the DFG Heisenberg Program.","has_accepted_license":"1","publisher":"Cambridge University Press","status":"public","date_updated":"2026-05-04T09:42:57Z","article_processing_charge":"Yes (via OA deal)","_id":"21706","page":"3089-3139","intvolume":" 161","department":[{"_id":"MaKw"}],"external_id":{"arxiv":["2312.13826"]},"volume":161,"year":"2025","oa_version":"Published Version","language":[{"iso":"eng"}],"corr_author":"1","abstract":[{"lang":"eng","text":"Consider a quadratic polynomial Q(ξ1, . . . , ξn) of independent Rademacher random variables ξ1, . . . , ξn. To what extent can Q(ξ1, . . . , ξn) concentrate on a single value? This quadratic version of the classical Littlewood–Offord problem was popularised by Costello, Tao and Vu in their study of symmetric random matrices. In this paper, we obtain an essentially optimal bound for this problem, as conjectured by Nguyen and Vu. Specifically, if Q(ξ1, . . . , ξn) ‘robustly depends on at least m of the ξi’ in the sense that there is no way to pin down the value of Q(ξ1, . . . , ξn) by fixing values for fewer than m of the variables ξi, then we have Pr[Q(ξ1, . . . , ξn) = 0] ≤ O(1/√m). This also implies a similar result in the case where ξ1, . . . , ξn have arbitrary distributions. Our proof combines a number of ideas that may be of independent interest, including an inductive decoupling scheme that reduces quadratic anticoncentration problems\r\nto high-dimensional linear anticoncentration problems. Also, one application of our main result is the resolution of a conjecture of Alon, Hefetz, Krivelevich and Tyomkyn related to graph inducibility. "}],"file":[{"access_level":"open_access","relation":"main_file","file_id":"21787","content_type":"application/pdf","checksum":"bd3415bb435da9d0b39f6f9a18c61abb","file_size":858727,"file_name":"2025_CompositioMath_Kwan.pdf","success":1,"creator":"dernst","date_updated":"2026-05-04T09:41:25Z","date_created":"2026-05-04T09:41:25Z"}],"project":[{"name":"Randomness and structure in combinatorics","grant_number":"101076777","_id":"bd95085b-d553-11ed-ba76-e55d3349be45"}],"quality_controlled":"1","issue":"12","publication":"Compositio Mathematica","title":"Resolution of the quadratic Littlewood–Offord problem","doi":"10.1112/S0010437X25102789","arxiv":1,"citation":{"ieee":"M. A. Kwan and L. Sauermann, “Resolution of the quadratic Littlewood–Offord problem,” Compositio Mathematica, vol. 161, no. 12. Cambridge University Press, pp. 3089–3139, 2025.","ama":"Kwan MA, Sauermann L. Resolution of the quadratic Littlewood–Offord problem. Compositio Mathematica. 2025;161(12):3089-3139. doi:10.1112/S0010437X25102789","ista":"Kwan MA, Sauermann L. 2025. Resolution of the quadratic Littlewood–Offord problem. Compositio Mathematica. 161(12), 3089–3139.","apa":"Kwan, M. A., & Sauermann, L. (2025). Resolution of the quadratic Littlewood–Offord problem. Compositio Mathematica. Cambridge University Press. https://doi.org/10.1112/S0010437X25102789","mla":"Kwan, Matthew Alan, and Lisa Sauermann. “Resolution of the Quadratic Littlewood–Offord Problem.” Compositio Mathematica, vol. 161, no. 12, Cambridge University Press, 2025, pp. 3089–139, doi:10.1112/S0010437X25102789.","short":"M.A. Kwan, L. Sauermann, Compositio Mathematica 161 (2025) 3089–3139.","chicago":"Kwan, Matthew Alan, and Lisa Sauermann. “Resolution of the Quadratic Littlewood–Offord Problem.” Compositio Mathematica. Cambridge University Press, 2025. https://doi.org/10.1112/S0010437X25102789."},"publication_identifier":{"issn":["0010-437X"],"eissn":["1570-5846"]},"date_created":"2026-04-12T22:01:48Z","month":"12","author":[{"full_name":"Kwan, Matthew Alan","id":"5fca0887-a1db-11eb-95d1-ca9d5e0453b3","last_name":"Kwan","orcid":"0000-0002-4003-7567","first_name":"Matthew Alan"},{"full_name":"Sauermann, Lisa","last_name":"Sauermann","first_name":"Lisa"}],"type":"journal_article","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)"},"OA_place":"publisher","PlanS_conform":"1","scopus_import":"1","file_date_updated":"2026-05-04T09:41:25Z","publication_status":"published","article_type":"original","date_published":"2025-12-01T00:00:00Z","fulldoi":"https://doi.org/10.1112/S0010437X25102789","OA_type":"hybrid","day":"01"},{"quality_controlled":"1","title":"On the chromatic number of powers of subdivisions of graphs","publication":"Discrete Applied Mathematics","arxiv":1,"publication_identifier":{"issn":["0166-218X"]},"citation":{"ieee":"M. Anastos, S. Boyadzhiyska, S. Rathke, and J. Rué, “On the chromatic number of powers of subdivisions of graphs,” Discrete Applied Mathematics, vol. 360. Elsevier, pp. 506–511, 2025.","ama":"Anastos M, Boyadzhiyska S, Rathke S, Rué J. On the chromatic number of powers of subdivisions of graphs. Discrete Applied Mathematics. 2025;360:506-511. doi:10.1016/j.dam.2024.10.002","ista":"Anastos M, Boyadzhiyska S, Rathke S, Rué J. 2025. On the chromatic number of powers of subdivisions of graphs. Discrete Applied Mathematics. 360, 506–511.","apa":"Anastos, M., Boyadzhiyska, S., Rathke, S., & Rué, J. (2025). On the chromatic number of powers of subdivisions of graphs. Discrete Applied Mathematics. Elsevier. https://doi.org/10.1016/j.dam.2024.10.002","mla":"Anastos, Michael, et al. “On the Chromatic Number of Powers of Subdivisions of Graphs.” Discrete Applied Mathematics, vol. 360, Elsevier, 2025, pp. 506–11, doi:10.1016/j.dam.2024.10.002.","short":"M. Anastos, S. Boyadzhiyska, S. Rathke, J. Rué, Discrete Applied Mathematics 360 (2025) 506–511.","chicago":"Anastos, Michael, Simona Boyadzhiyska, Silas Rathke, and Juanjo Rué. “On the Chromatic Number of Powers of Subdivisions of Graphs.” Discrete Applied Mathematics. Elsevier, 2025. https://doi.org/10.1016/j.dam.2024.10.002."},"doi":"10.1016/j.dam.2024.10.002","month":"01","author":[{"first_name":"Michael","last_name":"Anastos","id":"0b2a4358-bb35-11ec-b7b9-e3279b593dbb","full_name":"Anastos, Michael"},{"full_name":"Boyadzhiyska, Simona","last_name":"Boyadzhiyska","first_name":"Simona"},{"first_name":"Silas","last_name":"Rathke","full_name":"Rathke, Silas"},{"full_name":"Rué, Juanjo","last_name":"Rué","first_name":"Juanjo"}],"date_created":"2024-10-27T23:01:44Z","type":"journal_article","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)"},"ec_funded":1,"OA_place":"publisher","fulldoi":"https://doi.org/10.1016/j.dam.2024.10.002","day":"15","OA_type":"hybrid","scopus_import":"1","file_date_updated":"2025-01-13T09:25:59Z","article_type":"original","publication_status":"published","date_published":"2025-01-15T00:00:00Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","acknowledgement":"This work was initiated at the annual workshop of the Combinatorics and Graph Theory group of Freie Universität Berlin in Wilhelmsaue in September 2023. The authors would like to thank the institution for enabling this research. Finally, the fourth author would like to thank Tibor Szabó and the Combinatorics and Graph Theory group at Freie Universität Berlin for their hospitality during the research visit. Additionally, we thank Moharram Iradmusa for bringing the papers [5], [7] to our attention. Finally, we thank the anonymous referees for their suggestions on the manuscript, which have improved the quality of the document.\r\nM.A.: This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 101034413 .\r\nS.B.: The research leading to these results was supported by EPSRC, UK, grant no. EP/V048287/1. There are no additional data beyond that contained within the main manuscript.\r\nS.R.: Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – The Berlin Mathematics Research Center MATH+ (EXC-2046/1, project ID: 390685689).\r\nJ.R. acknowledges the support of the Grant PID2020-113082GB-I00 funded by MICIU/AEI/10.13039/501100011033, Spain, and the Severo Ochoa and María de Maeztu Program for Centers and Units of Excellence in R&D, Spain (CEX2020-001084-M).","oa":1,"ddc":["510"],"has_accepted_license":"1","publisher":"Elsevier","_id":"18478","page":"506-511","date_updated":"2025-04-14T07:54:56Z","status":"public","article_processing_charge":"Yes (in subscription journal)","department":[{"_id":"MaKw"}],"intvolume":" 360","oa_version":"Published Version","language":[{"iso":"eng"}],"external_id":{"arxiv":["2404.05542"],"isi":["001343647000001"]},"year":"2025","volume":360,"isi":1,"project":[{"name":"IST-BRIDGE: International postdoctoral program","grant_number":"101034413","_id":"fc2ed2f7-9c52-11eb-aca3-c01059dda49c","call_identifier":"H2020"}],"abstract":[{"text":"For a given graph G=(V,E), we define its \\emph{nth subdivision} as the graph obtained from G by replacing every edge by a path of length n. We also define the \\emph{mth power} of G as the graph on vertex set V where we connect every pair of vertices at distance at most m in G. In this paper, we study the chromatic number of powers of subdivisions of graphs and resolve the case m=n asymptotically. In particular, our result confirms a conjecture of Mozafari-Nia and Iradmusa in the case m=n=3 in a strong sense.","lang":"eng"}],"corr_author":"1","file":[{"checksum":"bd20a13e56b3ea01daf5e7aca5247c60","content_type":"application/pdf","file_id":"18836","relation":"main_file","access_level":"open_access","date_updated":"2025-01-13T09:25:59Z","date_created":"2025-01-13T09:25:59Z","creator":"dernst","success":1,"file_size":441060,"file_name":"2025_DiscreteApplMath_Anastos.pdf"}]},{"isi":1,"abstract":[{"text":"A k-subcolouring of a graph G is a function f : V (G) → {0,...,k − 1} such that the set of\r\nvertices coloured i induce a disjoint union of cliques. The subchromatic number, χsub(G),\r\nis the minimum k such that G admits a k-subcolouring. Nešetril, ˇ Ossona de Mendez,\r\nPilipczuk, and Zhu (2020), recently raised the problem of finding tight upper bounds for\r\nχsub(G2) when G is planar. We show that χsub(G2) ≤ 43 when G is planar, improving\r\ntheir bound of 135. We give even better bounds when the planar graph G has larger girth.\r\nMoreover, we show that χsub(G3) ≤ 95, improving the previous bound of 364. For these\r\nwe adapt some recent techniques of Almulhim and Kierstead (2022), while also extending\r\nthe decompositions of triangulated planar graphs of Van den Heuvel, Ossona de Mendez,\r\nQuiroz, Rabinovich and Siebertz (2017), to planar graphs of arbitrary girth. Note that these\r\ndecompositions are the precursors of the graph product structure theorem of planar graphs.\r\nWe give improved bounds for χsub(Gp) for all p ≥ 2, whenever G has bounded treewidth,\r\nbounded simple treewidth, bounded genus, or excludes a clique or biclique as a minor.\r\nFor this we introduce a family of parameters which form a gradation between the strong\r\nand the weak colouring numbers. We give upper bounds for these parameters for graphs\r\ncoming from such classes.\r\nFinally, we give a 2-approximation algorithm for the subchromatic number of graphs\r\nhaving a layering in which each layer has bounded cliquewidth and this layering is\r\ncomputable in polynomial time (like the class of all dth powers of planar graphs, for fixed\r\nd). This algorithm works even if the power p and the graph G is unknown.","lang":"eng"}],"corr_author":"1","file":[{"access_level":"open_access","relation":"main_file","content_type":"application/pdf","file_id":"19657","checksum":"6723cbb02b6aea0d05f37d167da00c03","success":1,"file_name":"2025_DiscreteMath_Cortes.pdf","file_size":850988,"creator":"dernst","date_updated":"2025-05-05T12:56:12Z","date_created":"2025-05-05T12:56:12Z"}],"oa_version":"Published Version","language":[{"iso":"eng"}],"external_id":{"arxiv":["2306.02195"],"isi":["001401656900001"]},"volume":348,"year":"2025","department":[{"_id":"MaKw"}],"intvolume":" 348","_id":"19002","status":"public","date_updated":"2025-09-30T10:25:15Z","article_processing_charge":"Yes (via OA deal)","has_accepted_license":"1","publisher":"Elsevier","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","acknowledgement":"We thank an anonymous referee for pointing out an error in an earlier version of Theorem 3.1. We also thank an anonymous referee for pointing out numerous typos in an earlier version of the paper.","oa":1,"ddc":["510"],"fulldoi":"https://doi.org/10.1016/j.disc.2024.114377","OA_type":"hybrid","day":"01","scopus_import":"1","date_published":"2025-04-01T00:00:00Z","publication_status":"published","file_date_updated":"2025-05-05T12:56:12Z","article_type":"original","OA_place":"publisher","type":"journal_article","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)"},"month":"04","article_number":"114377","author":[{"last_name":"Cortés","first_name":"Pedro P.","full_name":"Cortés, Pedro P."},{"full_name":"Kumar, Pankaj","last_name":"Kumar","first_name":"Pankaj"},{"full_name":"Moore, Benjamin","first_name":"Benjamin","id":"6dc1a1be-bf1c-11ed-8d2b-d044840f49d6","last_name":"Moore"},{"first_name":"Patrice","last_name":"Ossona de Mendez","full_name":"Ossona de Mendez, Patrice"},{"first_name":"Daniel A.","last_name":"Quiroz","full_name":"Quiroz, Daniel A."}],"date_created":"2025-02-05T06:51:08Z","arxiv":1,"citation":{"ieee":"P. P. Cortés, P. Kumar, B. Moore, P. Ossona de Mendez, and D. A. Quiroz, “Subchromatic numbers of powers of graphs with excluded minors,” Discrete Mathematics, vol. 348, no. 4. Elsevier, 2025.","mla":"Cortés, Pedro P., et al. “Subchromatic Numbers of Powers of Graphs with Excluded Minors.” Discrete Mathematics, vol. 348, no. 4, 114377, Elsevier, 2025, doi:10.1016/j.disc.2024.114377.","ama":"Cortés PP, Kumar P, Moore B, Ossona de Mendez P, Quiroz DA. Subchromatic numbers of powers of graphs with excluded minors. Discrete Mathematics. 2025;348(4). doi:10.1016/j.disc.2024.114377","apa":"Cortés, P. P., Kumar, P., Moore, B., Ossona de Mendez, P., & Quiroz, D. A. (2025). Subchromatic numbers of powers of graphs with excluded minors. Discrete Mathematics. Elsevier. https://doi.org/10.1016/j.disc.2024.114377","ista":"Cortés PP, Kumar P, Moore B, Ossona de Mendez P, Quiroz DA. 2025. Subchromatic numbers of powers of graphs with excluded minors. Discrete Mathematics. 348(4), 114377.","short":"P.P. Cortés, P. Kumar, B. Moore, P. Ossona de Mendez, D.A. Quiroz, Discrete Mathematics 348 (2025).","chicago":"Cortés, Pedro P., Pankaj Kumar, Benjamin Moore, Patrice Ossona de Mendez, and Daniel A. Quiroz. “Subchromatic Numbers of Powers of Graphs with Excluded Minors.” Discrete Mathematics. Elsevier, 2025. https://doi.org/10.1016/j.disc.2024.114377."},"publication_identifier":{"issn":["0012-365X"]},"doi":"10.1016/j.disc.2024.114377","title":"Subchromatic numbers of powers of graphs with excluded minors","publication":"Discrete Mathematics","issue":"4","quality_controlled":"1"},{"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)"},"type":"journal_article","author":[{"full_name":"Glock, Stefan","last_name":"Glock","first_name":"Stefan"},{"first_name":"Jaehoon","last_name":"Kim","full_name":"Kim, Jaehoon"},{"full_name":"Lichev, Lyuben","id":"9aa8388e-d003-11ee-8458-c4c1d7447977","last_name":"Lichev","first_name":"Lyuben"},{"last_name":"Pikhurko","first_name":"Oleg","full_name":"Pikhurko, Oleg"},{"full_name":"Sun, Shumin","first_name":"Shumin","last_name":"Sun"}],"month":"01","date_created":"2025-02-10T08:39:46Z","OA_type":"hybrid","day":"06","fulldoi":"https://doi.org/10.4153/s0008414x25000021","publication_status":"epub_ahead","date_published":"2025-01-06T00:00:00Z","article_type":"original","scopus_import":"1","OA_place":"publisher","quality_controlled":"1","citation":{"mla":"Glock, Stefan, et al. “On the (k + 2, k)-Problem of Brown, Erdős, and Sós for k = 5,6,7.” Canadian Journal of Mathematics, Cambridge University Press, 2025, pp. 1–43, doi:10.4153/s0008414x25000021.","ista":"Glock S, Kim J, Lichev L, Pikhurko O, Sun S. 2025. On the (k + 2, k)-problem of Brown, Erdős, and Sós for k = 5,6,7. Canadian Journal of Mathematics., 1–43.","ama":"Glock S, Kim J, Lichev L, Pikhurko O, Sun S. On the (k + 2, k)-problem of Brown, Erdős, and Sós for k = 5,6,7. Canadian Journal of Mathematics. 2025:1-43. doi:10.4153/s0008414x25000021","apa":"Glock, S., Kim, J., Lichev, L., Pikhurko, O., & Sun, S. (2025). On the (k + 2, k)-problem of Brown, Erdős, and Sós for k = 5,6,7. Canadian Journal of Mathematics. Cambridge University Press. https://doi.org/10.4153/s0008414x25000021","ieee":"S. Glock, J. Kim, L. Lichev, O. Pikhurko, and S. Sun, “On the (k + 2, k)-problem of Brown, Erdős, and Sós for k = 5,6,7,” Canadian Journal of Mathematics. Cambridge University Press, pp. 1–43, 2025.","chicago":"Glock, Stefan, Jaehoon Kim, Lyuben Lichev, Oleg Pikhurko, and Shumin Sun. “On the (k + 2, k)-Problem of Brown, Erdős, and Sós for k = 5,6,7.” Canadian Journal of Mathematics. Cambridge University Press, 2025. https://doi.org/10.4153/s0008414x25000021.","short":"S. Glock, J. Kim, L. Lichev, O. Pikhurko, S. Sun, Canadian Journal of Mathematics (2025) 1–43."},"publication_identifier":{"issn":["0008-414X"],"eissn":["1496-4279"]},"arxiv":1,"doi":"10.4153/s0008414x25000021","title":"On the (k + 2, k)-problem of Brown, Erdős, and Sós for k = 5,6,7","main_file_link":[{"open_access":"1","url":"https://doi.org/10.4153/s0008414x25000021"}],"publication":"Canadian Journal of Mathematics","department":[{"_id":"MaKw"}],"page":"1-43","_id":"19017","article_processing_charge":"No","status":"public","date_updated":"2025-09-30T10:28:07Z","isi":1,"abstract":[{"text":"Let f(r)(n;s,k) denote the maximum number of edges in an n-vertex r-uniform hypergraph containing no subgraph with k edges and at most s vertices. Brown, Erdős and Sós [New directions in the theory of graphs (Proc. Third Ann Arbor Conf., Univ. Michigan 1971), pp. 53--63, Academic Press 1973] conjectured that the limit limn→∞n−2f(3)(n;k+2,k) exists for all k. The value of the limit was previously determined for k=2 in the original paper of Brown, Erdős and Sós, for k=3 by Glock [Bull. Lond. Math. Soc. 51 (2019) 230--236] and for k=4 by Glock, Joos, Kim, Kühn, Lichev and Pikhurko [arXiv:2209.14177, accepted by Proc. Amer. Math. Soc.] while Delcourt and Postle [arXiv:2210.01105, accepted by Proc. Amer. Math. Soc.] proved the conjecture (without determining the limiting value).\r\nIn this paper, we determine the value of the limit in the Brown-Erdős-Sós Problem for k∈{5,6,7}. More generally, we obtain the value of limn→∞n−2f(r)(n;rk−2k+2,k) for all r≥3 and k∈{5,6,7}. In addition, by combining these new values with recent results of Bennett, Cushman and Dudek [arXiv:2309.00182] we obtain new asymptotic values for several generalised Ramsey numbers.","lang":"eng"}],"language":[{"iso":"eng"}],"oa_version":"Published Version","year":"2025","external_id":{"isi":["001416788600001"],"arxiv":["2403.04474"]},"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","oa":1,"ddc":["500"],"has_accepted_license":"1","publisher":"Cambridge University Press"},{"OA_place":"repository","publication_status":"published","date_published":"2025-05-01T00:00:00Z","article_type":"original","scopus_import":"1","OA_type":"green","day":"01","fulldoi":"https://doi.org/10.1016/j.ejc.2025.104120","date_created":"2025-02-10T09:00:53Z","author":[{"full_name":"Burova, Sofiya","first_name":"Sofiya","last_name":"Burova"},{"full_name":"Lichev, Lyuben","last_name":"Lichev","id":"9aa8388e-d003-11ee-8458-c4c1d7447977","first_name":"Lyuben"}],"article_number":"104120","month":"05","type":"journal_article","publication":"European Journal of Combinatorics","title":"The semi-random tree process","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2204.07376"}],"doi":"10.1016/j.ejc.2025.104120","publication_identifier":{"issn":["0195-6698"]},"citation":{"ieee":"S. Burova and L. Lichev, “The semi-random tree process,” European Journal of Combinatorics, vol. 126. Elsevier, 2025.","mla":"Burova, Sofiya, and Lyuben Lichev. “The Semi-Random Tree Process.” European Journal of Combinatorics, vol. 126, 104120, Elsevier, 2025, doi:10.1016/j.ejc.2025.104120.","ista":"Burova S, Lichev L. 2025. The semi-random tree process. European Journal of Combinatorics. 126, 104120.","apa":"Burova, S., & Lichev, L. (2025). The semi-random tree process. European Journal of Combinatorics. Elsevier. https://doi.org/10.1016/j.ejc.2025.104120","ama":"Burova S, Lichev L. The semi-random tree process. European Journal of Combinatorics. 2025;126. doi:10.1016/j.ejc.2025.104120","short":"S. Burova, L. Lichev, European Journal of Combinatorics 126 (2025).","chicago":"Burova, Sofiya, and Lyuben Lichev. “The Semi-Random Tree Process.” European Journal of Combinatorics. Elsevier, 2025. https://doi.org/10.1016/j.ejc.2025.104120."},"arxiv":1,"quality_controlled":"1","volume":126,"year":"2025","external_id":{"isi":["001420659400001"],"arxiv":["2204.07376 "]},"language":[{"iso":"eng"}],"oa_version":"Preprint","abstract":[{"lang":"eng","text":"The online semi-random graph process is a one-player game which starts with the empty graph on n vertices. At every round, a player (called Builder) is presented with a vertex v chosen uniformly at random and independently from previous rounds, and constructs an edge of their choice that is incident to v. Inspired by recent advances on the semi-random graph process, we define a family of generalized online semi-random models.\r\nWe analyse a particular instance that shares similar features with the original semi-random graph process and determine the hitting times of the classical graph properties minimum degree k,k-connectivity, containment of a perfect matching, a Hamiltonian cycle and an \r\nH-factor for a fixed graph H possessing an additional tree-like property. Along the way, we derive a few consequences of the famous Aldous-Broder algorithm that may be of independent interest."}],"isi":1,"article_processing_charge":"No","date_updated":"2025-09-30T10:28:42Z","status":"public","_id":"19018","intvolume":" 126","department":[{"_id":"MaKw"}],"publisher":"Elsevier","oa":1,"acknowledgement":"We are grateful to Dieter Mitsche for related discussions and to several anonymous referees for multiple useful comments.","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345"},{"scopus_import":"1","publication_status":"published","article_type":"original","file_date_updated":"2025-03-20T09:46:20Z","date_published":"2025-03-01T00:00:00Z","fulldoi":"https://doi.org/10.1112/jlms.70116","OA_type":"hybrid","day":"01","OA_place":"publisher","ec_funded":1,"type":"journal_article","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)"},"date_created":"2025-03-19T09:03:37Z","month":"03","article_number":"e70116","author":[{"full_name":"Draganić, Nemanja","last_name":"Draganić","first_name":"Nemanja"},{"last_name":"Petrova","id":"554ff4e4-f325-11ee-b0c4-a10dbd523381","first_name":"Kalina H","full_name":"Petrova, Kalina H"}],"doi":"10.1112/jlms.70116","arxiv":1,"citation":{"mla":"Draganić, Nemanja, and Kalina H. Petrova. “Size‐Ramsey Numbers of Graphs with Maximum Degree Three.” Journal of the London Mathematical Society, vol. 111, no. 3, e70116, Wiley, 2025, doi:10.1112/jlms.70116.","ista":"Draganić N, Petrova KH. 2025. Size‐Ramsey numbers of graphs with maximum degree three. Journal of the London Mathematical Society. 111(3), e70116.","apa":"Draganić, N., & Petrova, K. H. (2025). Size‐Ramsey numbers of graphs with maximum degree three. Journal of the London Mathematical Society. Wiley. https://doi.org/10.1112/jlms.70116","ama":"Draganić N, Petrova KH. Size‐Ramsey numbers of graphs with maximum degree three. Journal of the London Mathematical Society. 2025;111(3). doi:10.1112/jlms.70116","ieee":"N. Draganić and K. H. Petrova, “Size‐Ramsey numbers of graphs with maximum degree three,” Journal of the London Mathematical Society, vol. 111, no. 3. Wiley, 2025.","chicago":"Draganić, Nemanja, and Kalina H Petrova. “Size‐Ramsey Numbers of Graphs with Maximum Degree Three.” Journal of the London Mathematical Society. Wiley, 2025. https://doi.org/10.1112/jlms.70116.","short":"N. Draganić, K.H. Petrova, Journal of the London Mathematical Society 111 (2025)."},"publication_identifier":{"eissn":["1469-7750"],"issn":["0024-6107"]},"publication":"Journal of the London Mathematical Society","title":"Size‐Ramsey numbers of graphs with maximum degree three","quality_controlled":"1","issue":"3","abstract":[{"text":"The size-Ramsey number r^(H) of a graph H is the smallest number of edges a (host) graph G can have, such that for any red/blue colouring of G, there is a monochromatic copy of H in G. Recently, Conlon, Nenadov and Trujić showed that if H is a graph on n vertices and maximum degree three, then r^(H)=O(n8/5), improving upon the upper bound of n5/3+o(1) by Kohayakawa, Rödl, Schacht and Szemerédi. In this paper we show that r^(H)≤n3/2+o(1). While the previously used host graphs were vanilla binomial random graphs, we prove our result using a novel host graph construction. Our bound hits a natural barrier of the existing methods.","lang":"eng"}],"file":[{"creator":"dernst","date_updated":"2025-03-20T09:46:20Z","date_created":"2025-03-20T09:46:20Z","file_size":625974,"file_name":"2025_JournLondonMath_Draganic.pdf","success":1,"content_type":"application/pdf","file_id":"19427","checksum":"d8e0a03286a44c4f672709e3c829206e","access_level":"open_access","relation":"main_file"}],"isi":1,"project":[{"call_identifier":"H2020","_id":"fc2ed2f7-9c52-11eb-aca3-c01059dda49c","grant_number":"101034413","name":"IST-BRIDGE: International postdoctoral program"}],"external_id":{"arxiv":["2207.05048"],"isi":["001450645400019"]},"year":"2025","volume":111,"oa_version":"Published Version","language":[{"iso":"eng"}],"intvolume":" 111","department":[{"_id":"MaKw"}],"date_updated":"2025-09-30T11:05:07Z","status":"public","article_processing_charge":"No","_id":"19418","publisher":"Wiley","has_accepted_license":"1","oa":1,"ddc":["510"],"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","acknowledgement":"We would like to thank Rajko Nenadov and Miloš Trujić for helpful comments and discussions, as well as the anonymous referees for their very useful feedback, which improved the paper considerably. This research was supported by SNSF Project 217926. Part of this research was conducted while Nemanja Draganić was at ETH Zürich, Switzerland, and partially supported by SNSF Grant 200021_196965. This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement Number: 101034413. Part of this research was conducted while Kalina Petrova was at the Department of Computer Science, ETH Zürich, Switzerland, supported by SNSF Grant CRSII5 173721."},{"acknowledgement":"We would like to thank Timo Seppäläinen for some illuminating discussion about random high-dimensional orders and for bringing our attention to [59]. We would also like to thank the referees for helpful feedback. Michael Anastos is supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 101034413. Matthew Kwan is supported by ERC Starting Grant ‘RANDSTRUCT’ No. 101076777, also funded by the European Union. Zhihan Jin and Benny Sudakov are supported by SNSF grant 200021-228014.","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","ddc":["510"],"oa":1,"publisher":"Cambridge University Press","has_accepted_license":"1","department":[{"_id":"MaKw"}],"intvolume":" 13","_id":"19433","article_processing_charge":"Yes","status":"public","date_updated":"2025-09-30T11:18:57Z","project":[{"call_identifier":"H2020","_id":"fc2ed2f7-9c52-11eb-aca3-c01059dda49c","grant_number":"101034413","name":"IST-BRIDGE: International postdoctoral program"},{"_id":"bd95085b-d553-11ed-ba76-e55d3349be45","grant_number":"101076777","name":"Randomness and structure in combinatorics"}],"isi":1,"file":[{"relation":"main_file","access_level":"open_access","checksum":"f396270ad78c1ed67095c8e5a66fca26","content_type":"application/pdf","file_id":"19468","success":1,"file_name":"2025_ForumMathSigma_Anastos.pdf","file_size":630297,"date_created":"2025-04-03T11:24:35Z","date_updated":"2025-04-03T11:24:35Z","creator":"dernst"}],"abstract":[{"lang":"eng","text":"An ordered r-matching is an r-uniform hypergraph matching equipped with an ordering on its vertices. These objects can be viewed as natural generalisations of r-dimensional orders. The theory of ordered 2-matchings is well developed and has connections and applications to extremal and enumerative combinatorics, probability and geometry. On the other hand, in the case r≥3 much less is known, largely due to a lack of powerful bijective tools. Recently, Dudek, Grytczuk and Ruciński made some first steps towards a general theory of ordered r-matchings, and in this paper we substantially improve several of their results and introduce some new directions of study. Many intriguing open questions remain."}],"corr_author":"1","language":[{"iso":"eng"}],"oa_version":"Published Version","year":"2025","volume":13,"external_id":{"arxiv":["2308.12268"],"isi":["001444429200001"]},"quality_controlled":"1","citation":{"chicago":"Anastos, Michael, Zhihan Jin, Matthew Alan Kwan, and Benny Sudakov. “Extremal, Enumerative and Probabilistic Results on Ordered Hypergraph Matchings.” Forum of Mathematics, Sigma. Cambridge University Press, 2025. https://doi.org/10.1017/fms.2024.144.","short":"M. Anastos, Z. Jin, M.A. Kwan, B. Sudakov, Forum of Mathematics, Sigma 13 (2025).","mla":"Anastos, Michael, et al. “Extremal, Enumerative and Probabilistic Results on Ordered Hypergraph Matchings.” Forum of Mathematics, Sigma, vol. 13, e55, Cambridge University Press, 2025, doi:10.1017/fms.2024.144.","apa":"Anastos, M., Jin, Z., Kwan, M. A., & Sudakov, B. (2025). Extremal, enumerative and probabilistic results on ordered hypergraph matchings. Forum of Mathematics, Sigma. Cambridge University Press. https://doi.org/10.1017/fms.2024.144","ama":"Anastos M, Jin Z, Kwan MA, Sudakov B. Extremal, enumerative and probabilistic results on ordered hypergraph matchings. Forum of Mathematics, Sigma. 2025;13. doi:10.1017/fms.2024.144","ista":"Anastos M, Jin Z, Kwan MA, Sudakov B. 2025. Extremal, enumerative and probabilistic results on ordered hypergraph matchings. Forum of Mathematics, Sigma. 13, e55.","ieee":"M. Anastos, Z. Jin, M. A. Kwan, and B. Sudakov, “Extremal, enumerative and probabilistic results on ordered hypergraph matchings,” Forum of Mathematics, Sigma, vol. 13. Cambridge University Press, 2025."},"publication_identifier":{"issn":["2050-5094"]},"arxiv":1,"doi":"10.1017/fms.2024.144","title":"Extremal, enumerative and probabilistic results on ordered hypergraph matchings","publication":"Forum of Mathematics, Sigma","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)"},"type":"journal_article","ec_funded":1,"author":[{"full_name":"Anastos, Michael","last_name":"Anastos","id":"0b2a4358-bb35-11ec-b7b9-e3279b593dbb","first_name":"Michael"},{"first_name":"Zhihan","last_name":"Jin","full_name":"Jin, Zhihan"},{"id":"5fca0887-a1db-11eb-95d1-ca9d5e0453b3","last_name":"Kwan","first_name":"Matthew Alan","orcid":"0000-0002-4003-7567","full_name":"Kwan, Matthew Alan"},{"full_name":"Sudakov, Benny","last_name":"Sudakov","first_name":"Benny"}],"month":"03","article_number":"e55","date_created":"2025-03-20T12:59:14Z","OA_type":"gold","day":"14","fulldoi":"https://doi.org/10.1017/fms.2024.144","publication_status":"published","date_published":"2025-03-14T00:00:00Z","article_type":"original","file_date_updated":"2025-04-03T11:24:35Z","scopus_import":"1","OA_place":"publisher"},{"tmp":{"short":"CC BY-NC (4.0)","legal_code_url":"https://creativecommons.org/licenses/by-nc/4.0/legalcode","name":"Creative Commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0)","image":"/images/cc_by_nc.png"},"type":"journal_article","ec_funded":1,"author":[{"first_name":"Yahav","last_name":"Alon","full_name":"Alon, Yahav"},{"full_name":"Anastos, Michael","first_name":"Michael","last_name":"Anastos","id":"0b2a4358-bb35-11ec-b7b9-e3279b593dbb"}],"article_number":"e21286","month":"03","date_created":"2025-03-23T23:01:26Z","day":"01","OA_type":"hybrid","fulldoi":"https://doi.org/10.1002/rsa.21286","date_published":"2025-03-01T00:00:00Z","article_type":"original","file_date_updated":"2025-03-25T11:46:27Z","publication_status":"published","scopus_import":"1","OA_place":"publisher","issue":"2","quality_controlled":"1","publication_identifier":{"issn":["1042-9832"],"eissn":["1098-2418"]},"citation":{"ieee":"Y. Alon and M. Anastos, “The completion numbers of hamiltonicity and pancyclicity in random graphs,” Random Structures and Algorithms, vol. 66, no. 2. Wiley, 2025.","ista":"Alon Y, Anastos M. 2025. The completion numbers of hamiltonicity and pancyclicity in random graphs. Random Structures and Algorithms. 66(2), e21286.","ama":"Alon Y, Anastos M. The completion numbers of hamiltonicity and pancyclicity in random graphs. Random Structures and Algorithms. 2025;66(2). doi:10.1002/rsa.21286","apa":"Alon, Y., & Anastos, M. (2025). The completion numbers of hamiltonicity and pancyclicity in random graphs. Random Structures and Algorithms. Wiley. https://doi.org/10.1002/rsa.21286","mla":"Alon, Yahav, and Michael Anastos. “The Completion Numbers of Hamiltonicity and Pancyclicity in Random Graphs.” Random Structures and Algorithms, vol. 66, no. 2, e21286, Wiley, 2025, doi:10.1002/rsa.21286.","short":"Y. Alon, M. Anastos, Random Structures and Algorithms 66 (2025).","chicago":"Alon, Yahav, and Michael Anastos. “The Completion Numbers of Hamiltonicity and Pancyclicity in Random Graphs.” Random Structures and Algorithms. Wiley, 2025. https://doi.org/10.1002/rsa.21286."},"arxiv":1,"doi":"10.1002/rsa.21286","title":"The completion numbers of hamiltonicity and pancyclicity in random graphs","license":"https://creativecommons.org/licenses/by-nc/4.0/","publication":"Random Structures and Algorithms","department":[{"_id":"MaKw"}],"intvolume":" 66","_id":"19440","article_processing_charge":"Yes (in subscription journal)","status":"public","date_updated":"2025-09-30T11:15:41Z","isi":1,"project":[{"call_identifier":"H2020","name":"IST-BRIDGE: International postdoctoral program","grant_number":"101034413","_id":"fc2ed2f7-9c52-11eb-aca3-c01059dda49c"}],"file":[{"file_name":"2025_RandomStruc_Alon.pdf","file_size":549236,"success":1,"creator":"dernst","date_created":"2025-03-25T11:46:27Z","date_updated":"2025-03-25T11:46:27Z","access_level":"open_access","relation":"main_file","file_id":"19459","content_type":"application/pdf","checksum":"6067747e805fa356d560dc45f2a89918"}],"abstract":[{"lang":"eng","text":"Let μ(G) denote the minimum number of edges whose addition to G results in a Hamiltonian graph, and let μ^(G) denote the minimum number of edges whose addition to G results in a pancyclic graph. We study the distributions of μ(G),μ^(G) in the context of binomial random graphs. Letting d=d(n):=n⋅p, we prove that there exists a function f:R+→[0,1] of order f(d)=12de−d+e−d+O(d6e−3d) such that, if G∼G(n,p) with 20≤d(n)≤0.4logn, then with high probability μ(G)=(1+o(1))⋅f(d)⋅n. Let ni(G) denote the number of degree i vertices in G. A trivial lower bound on μ(G) is given by the expression n0(G)+⌈12n1(G)⌉. In the denser regime of random graphs, we show that if np−13logn−2loglogn→∞ and G∼G(n,p) then, with high probability, μ(G)=n0(G)+⌈12n1(G)⌉. For completion to pancyclicity, we show that if G∼G(n,p) and np≥20 then, with high probability, μ^(G)=μ(G). Finally, we present a polynomial time algorithm such that, if G∼G(n,p) and np≥20, then, with high probability, the algorithm returns a set of edges of size μ(G) whose addition to G results in a pancyclic (and therefore also Hamiltonian) graph."}],"language":[{"iso":"eng"}],"oa_version":"Published Version","volume":66,"year":"2025","external_id":{"arxiv":["2304.03710"],"isi":["001420226800001"]},"acknowledgement":"The authors would like to express their thanks to the referees of the article for their valuable input towards improving the presentation of our result. This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 101034413.","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","ddc":["510"],"oa":1,"has_accepted_license":"1","publisher":"Wiley"}]