[{"title":"A short proof of a central limit theorem for the order of the giant component and k-core","_id":"22752","author":[{"id":"0b2a4358-bb35-11ec-b7b9-e3279b593dbb","last_name":"Anastos","first_name":"Michael","full_name":"Anastos, Michael"},{"full_name":"Erde, Joshua","first_name":"Joshua","last_name":"Erde"},{"full_name":"Kang, Mihyun","last_name":"Kang","first_name":"Mihyun"},{"first_name":"Vincent","last_name":"Pfenninger","full_name":"Pfenninger, Vincent"}],"status":"public","date_created":"2026-08-23T22:01:47Z","type":"journal_article","oa_version":"Published Version","article_type":"original","abstract":[{"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.","lang":"eng"}],"citation":{"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.” <i>Bulletin of the London Mathematical Society</i>. Wiley, 2026. <a href=\"https://doi.org/10.1112/blms.70464\">https://doi.org/10.1112/blms.70464</a>.","apa":"Anastos, M., Erde, J., Kang, M., &#38; Pfenninger, V. (2026). A short proof of a central limit theorem for the order of the giant component and k-core. <i>Bulletin of the London Mathematical Society</i>. Wiley. <a href=\"https://doi.org/10.1112/blms.70464\">https://doi.org/10.1112/blms.70464</a>","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. <i>Bulletin of the London Mathematical Society</i>. 2026;58(8). doi:<a href=\"https://doi.org/10.1112/blms.70464\">10.1112/blms.70464</a>","mla":"Anastos, Michael, et al. “A Short Proof of a Central Limit Theorem for the Order of the Giant Component and K-Core.” <i>Bulletin of the London Mathematical Society</i>, vol. 58, no. 8, e70464, Wiley, 2026, doi:<a href=\"https://doi.org/10.1112/blms.70464\">10.1112/blms.70464</a>.","short":"M. Anastos, J. Erde, M. Kang, V. Pfenninger, Bulletin of the London Mathematical Society 58 (2026).","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.","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,” <i>Bulletin of the London Mathematical Society</i>, vol. 58, no. 8. Wiley, 2026."},"project":[{"grant_number":"ESP3863424","name":"Combinatorial Optimisation Problems on Sparse Random Graphs","_id":"8f906bd2-16d5-11f0-9cad-e07be8aa9ac9"}],"date_updated":"2026-09-07T13:56:17Z","supplementarymaterial":"no","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","PlanS_conform":"1","publication":"Bulletin of the London Mathematical Society","scopus_import":"1","issue":"8","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.","publication_identifier":{"issn":["0024-6093"],"eissn":["1469-2120"]},"OA_place":"publisher","OA_type":"hybrid","language":[{"iso":"eng"}],"tmp":{"image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"ddc":["510"],"publication_status":"published","quality_controlled":"1","year":"2026","intvolume":"        58","has_accepted_license":"1","file_date_updated":"2026-09-07T13:53:26Z","date_published":"2026-08-01T00:00:00Z","article_processing_charge":"Yes (in subscription journal)","das_tickbox":"0","fulldoi":"https://doi.org/10.1112/blms.70464","arxiv":1,"day":"01","oa":1,"external_id":{"arxiv":["2506.11651"]},"publisher":"Wiley","department":[{"_id":"MaKw"}],"mathsc":["05C80","60F05"],"month":"08","article_number":"e70464","volume":58,"researchdata_availability":"no","doi":"10.1112/blms.70464","file":[{"relation":"main_file","creator":"dernst","access_level":"open_access","checksum":"b4b21f05c3fc62fde243a5ddd46cef49","date_created":"2026-09-07T13:53:26Z","success":1,"file_size":248641,"file_name":"2026_BulletinLondonMathSoc_Anastos.pdf","date_updated":"2026-09-07T13:53:26Z","content_type":"application/pdf","file_id":"22845"}]},{"has_accepted_license":"1","date_published":"2026-10-01T00:00:00Z","file_date_updated":"2026-10-19T07:33:58Z","article_processing_charge":"Yes (in subscription journal)","keyword":["Random regular graphs","coupling","contiguity","total variation distance","decomposition"],"das_tickbox":"0","fulldoi":"https://doi.org/10.1214/26-AAP2345","arxiv":1,"ddc":["500"],"tmp":{"image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"publication_status":"published","quality_controlled":"1","year":"2026","intvolume":"        36","volume":36,"researchdata_availability":"no","doi":"10.1214/26-AAP2345","file":[{"success":1,"checksum":"46101e61d29cf40f0a204801d15a848f","date_created":"2026-10-19T07:33:58Z","access_level":"open_access","creator":"dernst","relation":"main_file","file_id":"23165","date_updated":"2026-10-19T07:33:58Z","content_type":"application/pdf","file_name":"2026_AnnalsAppliedProbability_Hollom.pdf","file_size":320593}],"day":"01","oa":1,"external_id":{"arxiv":["2505.22875"]},"publisher":"Institute of Mathematical Statistics","month":"10","department":[{"_id":"MaKw"},{"_id":"GradSch"}],"mathsc":["05C80"],"citation":{"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.","ieee":"L. Hollom, L. Lichev, A. Mond, J. Portier, and Y. Wang, “Monotonicity and decompositions of random regular graphs,” <i>Annals of Applied Probability</i>, vol. 36, no. 5. Institute of Mathematical Statistics, pp. 4716–4738, 2026.","short":"L. Hollom, L. Lichev, A. Mond, J. Portier, Y. Wang, Annals of Applied Probability 36 (2026) 4716–4738.","ama":"Hollom L, Lichev L, Mond A, Portier J, Wang Y. Monotonicity and decompositions of random regular graphs. <i>Annals of Applied Probability</i>. 2026;36(5):4716-4738. doi:<a href=\"https://doi.org/10.1214/26-AAP2345\">10.1214/26-AAP2345</a>","mla":"Hollom, Lawrence, et al. “Monotonicity and Decompositions of Random Regular Graphs.” <i>Annals of Applied Probability</i>, vol. 36, no. 5, Institute of Mathematical Statistics, 2026, pp. 4716–38, doi:<a href=\"https://doi.org/10.1214/26-AAP2345\">10.1214/26-AAP2345</a>.","apa":"Hollom, L., Lichev, L., Mond, A., Portier, J., &#38; Wang, Y. (2026). Monotonicity and decompositions of random regular graphs. <i>Annals of Applied Probability</i>. Institute of Mathematical Statistics. <a href=\"https://doi.org/10.1214/26-AAP2345\">https://doi.org/10.1214/26-AAP2345</a>","chicago":"Hollom, Lawrence, Lyuben Lichev, Adva Mond, Julien Portier, and Yiting Wang. “Monotonicity and Decompositions of Random Regular Graphs.” <i>Annals of Applied Probability</i>. Institute of Mathematical Statistics, 2026. <a href=\"https://doi.org/10.1214/26-AAP2345\">https://doi.org/10.1214/26-AAP2345</a>."},"page":"4716-4738","project":[{"_id":"bd95085b-d553-11ed-ba76-e55d3349be45","name":"Randomness and structure in combinatorics","grant_number":"101076777"},{"_id":"e19fef4c-1f3d-11f0-ac04-b8b1af6886d8","name":"THRESHOLDS IN SEMI-RANDOM PROCESSES AND GRAPH EMBEDDINGS","grant_number":"E624"}],"date_updated":"2026-10-19T07:39:17Z","supplementarymaterial":"no","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","PlanS_conform":"1","title":"Monotonicity and decompositions of random regular graphs","_id":"23155","status":"public","date_created":"2026-10-18T22:01:52Z","author":[{"full_name":"Hollom, Lawrence","last_name":"Hollom","first_name":"Lawrence"},{"last_name":"Lichev","first_name":"Lyuben","id":"9aa8388e-d003-11ee-8458-c4c1d7447977","full_name":"Lichev, Lyuben"},{"full_name":"Mond, Adva","first_name":"Adva","last_name":"Mond"},{"first_name":"Julien","last_name":"Portier","full_name":"Portier, Julien"},{"full_name":"Wang, Yiting","id":"1917d194-076e-11ed-97cd-837255f88785","first_name":"Yiting","last_name":"Wang"}],"type":"journal_article","oa_version":"Published Version","article_type":"original","abstract":[{"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.","lang":"eng"}],"OA_type":"hybrid","OA_place":"publisher","language":[{"iso":"eng"}],"publication":"Annals of Applied Probability","scopus_import":"1","issue":"5","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.","publication_identifier":{"issn":["1050-5164"],"eissn":["2168-8737"]}},{"status":"public","date_created":"2026-06-29T10:50:35Z","author":[{"full_name":"Christoph, Micha","first_name":"Micha","last_name":"Christoph"},{"full_name":"Martinsson, Anders","first_name":"Anders","last_name":"Martinsson"},{"full_name":"Steiner, Raphael","last_name":"Steiner","first_name":"Raphael"},{"full_name":"Wigderson, Yuval","id":"2d0023a0-1567-11f0-833d-d5c1e476d4b5","last_name":"Wigderson","first_name":"Yuval"}],"title":"Resolution of the Kohayakawa–Kreuter conjecture","_id":"22157","extern":"1","abstract":[{"text":"A graph 𝐺 is said to be Ramsey for a tuple of graphs(𝐻 1 , … , 𝐻𝑟 ) if every 𝑟-coloring of the edges of 𝐺 con-tains a monochromatic copy of 𝐻𝑖 in color 𝑖, for some 𝑖.A fundamental question at the intersection of Ramseytheory and the theory of random graphs is to deter-mine the threshold at which the binomial randomgraph 𝐺𝑛,𝑝 becomes asymptotically almost surely Ram-sey for a fixed tuple (𝐻 1 , … , 𝐻𝑟 ), and a famous conjectureof Kohayakawa and Kreuter predicts this threshold.Earlier work of Mousset–Nenadov–Samotij, Bowtell–Hancock–Hyde, and Kuperwasser–Samotij–Wigdersonhas reduced this probabilistic problem to a determinis-tic graph decomposition conjecture. In this paper, weresolve this deterministic problem, thus proving theKohayakawa–Kreuter conjecture. Along the way, weprove a number of novel graph decomposition resultsthat may be of independent interest.","lang":"eng"}],"oa_version":"Preprint","type":"journal_article","article_type":"original","citation":{"chicago":"Christoph, Micha, Anders Martinsson, Raphael Steiner, and Yuval Wigderson. “Resolution of the Kohayakawa–Kreuter Conjecture.” <i>Proceedings of the London Mathematical Society</i>. Wiley, 2025. <a href=\"https://doi.org/10.1112/plms.70013\">https://doi.org/10.1112/plms.70013</a>.","apa":"Christoph, M., Martinsson, A., Steiner, R., &#38; Wigderson, Y. (2025). Resolution of the Kohayakawa–Kreuter conjecture. <i>Proceedings of the London Mathematical Society</i>. Wiley. <a href=\"https://doi.org/10.1112/plms.70013\">https://doi.org/10.1112/plms.70013</a>","ama":"Christoph M, Martinsson A, Steiner R, Wigderson Y. Resolution of the Kohayakawa–Kreuter conjecture. <i>Proceedings of the London Mathematical Society</i>. 2025;130(1). doi:<a href=\"https://doi.org/10.1112/plms.70013\">10.1112/plms.70013</a>","mla":"Christoph, Micha, et al. “Resolution of the Kohayakawa–Kreuter Conjecture.” <i>Proceedings of the London Mathematical Society</i>, vol. 130, no. 1, e70013, Wiley, 2025, doi:<a href=\"https://doi.org/10.1112/plms.70013\">10.1112/plms.70013</a>.","ista":"Christoph M, Martinsson A, Steiner R, Wigderson Y. 2025. Resolution of the Kohayakawa–Kreuter conjecture. Proceedings of the London Mathematical Society. 130(1), e70013.","short":"M. Christoph, A. Martinsson, R. Steiner, Y. Wigderson, Proceedings of the London Mathematical Society 130 (2025).","ieee":"M. Christoph, A. Martinsson, R. Steiner, and Y. Wigderson, “Resolution of the Kohayakawa–Kreuter conjecture,” <i>Proceedings of the London Mathematical Society</i>, vol. 130, no. 1. Wiley, 2025."},"date_updated":"2026-07-08T10:24:21Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","scopus_import":"1","issue":"1","publication":"Proceedings of the London Mathematical Society","publication_identifier":{"eissn":["1460-244X"],"issn":["0024-6115"]},"OA_place":"repository","OA_type":"green","language":[{"iso":"eng"}],"ddc":["500"],"tmp":{"image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"intvolume":"       130","publication_status":"published","quality_controlled":"1","year":"2025","article_processing_charge":"No","date_published":"2025-01-01T00:00:00Z","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2402.03045","open_access":"1"}],"arxiv":1,"fulldoi":"https://doi.org/10.1112/plms.70013","oa":1,"day":"01","publisher":"Wiley","month":"01","mathsc":["05C70","05D10","05C80"],"external_id":{"arxiv":["2402.03045"]},"article_number":"e70013","doi":"10.1112/plms.70013","volume":130},{"day":"28","oa":1,"external_id":{"arxiv":["2307.16611"]},"publisher":"Cambridge University Press","mathsc":["05C80","05C55","05D10"],"month":"04","volume":178,"doi":"10.1017/s0305004125000143","publication_status":"published","year":"2025","quality_controlled":"1","intvolume":"       178","date_published":"2025-04-28T00:00:00Z","article_processing_charge":"No","fulldoi":"https://doi.org/10.1017/s0305004125000143","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2307.16611"}],"arxiv":1,"publication":"Mathematical Proceedings of the Cambridge Philosophical Society","scopus_import":"1","issue":"3","publication_identifier":{"eissn":["1469-8064"],"issn":["0305-0041"]},"OA_place":"repository","OA_type":"green","language":[{"iso":"eng"}],"title":"On the Kohayakawa–Kreuter conjecture","_id":"22168","status":"public","date_created":"2026-06-29T10:55:00Z","author":[{"full_name":"KUPERWASSER, EDEN","first_name":"EDEN","last_name":"KUPERWASSER"},{"first_name":"WOJCIECH","last_name":"SAMOTIJ","full_name":"SAMOTIJ, WOJCIECH"},{"id":"2d0023a0-1567-11f0-833d-d5c1e476d4b5","first_name":"Yuval","last_name":"Wigderson","full_name":"Wigderson, Yuval"}],"type":"journal_article","oa_version":"Preprint","article_type":"original","extern":"1","abstract":[{"lang":"eng","text":"Let us say that a graph G is Ramsey for a tuple (H1, ... , Hr) of graphs if every r-colouring\r\nof the edges of G contains a monochromatic copy of Hi in colour i, for some i ∈ [[r]].\r\nA famous conjecture of Kohayakawa and Kreuter, extending seminal work of Rödl and\r\nRucinski, predicts the threshold at which the binomial random graph ´ Gn,p becomes Ramsey\r\nfor (H1, ... , Hr) asymptotically almost surely.\r\nIn this paper, we resolve the Kohayakawa–Kreuter conjecture for almost all tuples of\r\ngraphs. Moreover, we reduce its validity to the truth of a certain deterministic statement,\r\nwhich is a clear necessary condition for the conjecture to hold. All of our results actually hold in greater generality, when one replaces the graphs H1, ... , Hr by finite families\r\nH1, ... , Hr. Additionally, we pose a natural (deterministic) graph-partitioning conjecture,\r\nwhich we believe to be of independent interest, and whose resolution would imply the\r\nKohayakawa–Kreuter conjecture."}],"page":"293-320","citation":{"chicago":"KUPERWASSER, EDEN, WOJCIECH SAMOTIJ, and Yuval Wigderson. “On the Kohayakawa–Kreuter Conjecture.” <i>Mathematical Proceedings of the Cambridge Philosophical Society</i>. Cambridge University Press, 2025. <a href=\"https://doi.org/10.1017/s0305004125000143\">https://doi.org/10.1017/s0305004125000143</a>.","apa":"KUPERWASSER, E., SAMOTIJ, W., &#38; Wigderson, Y. (2025). On the Kohayakawa–Kreuter conjecture. <i>Mathematical Proceedings of the Cambridge Philosophical Society</i>. Cambridge University Press. <a href=\"https://doi.org/10.1017/s0305004125000143\">https://doi.org/10.1017/s0305004125000143</a>","ama":"KUPERWASSER E, SAMOTIJ W, Wigderson Y. On the Kohayakawa–Kreuter conjecture. <i>Mathematical Proceedings of the Cambridge Philosophical Society</i>. 2025;178(3):293-320. doi:<a href=\"https://doi.org/10.1017/s0305004125000143\">10.1017/s0305004125000143</a>","mla":"KUPERWASSER, EDEN, et al. “On the Kohayakawa–Kreuter Conjecture.” <i>Mathematical Proceedings of the Cambridge Philosophical Society</i>, vol. 178, no. 3, Cambridge University Press, 2025, pp. 293–320, doi:<a href=\"https://doi.org/10.1017/s0305004125000143\">10.1017/s0305004125000143</a>.","ieee":"E. KUPERWASSER, W. SAMOTIJ, and Y. Wigderson, “On the Kohayakawa–Kreuter conjecture,” <i>Mathematical Proceedings of the Cambridge Philosophical Society</i>, vol. 178, no. 3. Cambridge University Press, pp. 293–320, 2025.","short":"E. KUPERWASSER, W. SAMOTIJ, Y. Wigderson, Mathematical Proceedings of the Cambridge Philosophical Society 178 (2025) 293–320.","ista":"KUPERWASSER E, SAMOTIJ W, Wigderson Y. 2025. On the Kohayakawa–Kreuter conjecture. Mathematical Proceedings of the Cambridge Philosophical Society. 178(3), 293–320."},"date_updated":"2026-07-14T08:38:08Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87"}]