{"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."}],"OA_type":"hybrid","article_processing_charge":"Yes (in subscription journal)","scopus_import":"1","file":[{"checksum":"6067747e805fa356d560dc45f2a89918","relation":"main_file","file_size":549236,"access_level":"open_access","file_name":"2025_RandomStruc_Alon.pdf","content_type":"application/pdf","creator":"dernst","success":1,"file_id":"19459","date_created":"2025-03-25T11:46:27Z","date_updated":"2025-03-25T11:46:27Z"}],"author":[{"last_name":"Alon","full_name":"Alon, Yahav","first_name":"Yahav"},{"full_name":"Anastos, Michael","first_name":"Michael","last_name":"Anastos","id":"0b2a4358-bb35-11ec-b7b9-e3279b593dbb"}],"oa":1,"external_id":{"arxiv":["2304.03710"]},"publication_status":"published","volume":66,"article_number":"e21286","publisher":"Wiley","tmp":{"name":"Creative Commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0)","image":"/images/cc_by_nc.png","short":"CC BY-NC (4.0)","legal_code_url":"https://creativecommons.org/licenses/by-nc/4.0/legalcode"},"publication_identifier":{"eissn":["1098-2418"],"issn":["1042-9832"]},"file_date_updated":"2025-03-25T11:46:27Z","doi":"10.1002/rsa.21286","license":"https://creativecommons.org/licenses/by-nc/4.0/","date_updated":"2025-03-25T11:48:52Z","department":[{"_id":"MaKw"}],"month":"03","title":"The completion numbers of hamiltonicity and pancyclicity in random graphs","day":"01","language":[{"iso":"eng"}],"year":"2025","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.","publication":"Random Structures and Algorithms","date_published":"2025-03-01T00:00:00Z","arxiv":1,"OA_place":"publisher","project":[{"_id":"fc2ed2f7-9c52-11eb-aca3-c01059dda49c","call_identifier":"H2020","grant_number":"101034413","name":"IST-BRIDGE: International postdoctoral program"}],"quality_controlled":"1","status":"public","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","ddc":["510"],"type":"journal_article","has_accepted_license":"1","date_created":"2025-03-23T23:01:26Z","article_type":"original","citation":{"apa":"Alon, Y., &#38; Anastos, M. (2025). The completion numbers of hamiltonicity and pancyclicity in random graphs. <i>Random Structures and Algorithms</i>. Wiley. <a href=\"https://doi.org/10.1002/rsa.21286\">https://doi.org/10.1002/rsa.21286</a>","short":"Y. Alon, M. Anastos, Random Structures and Algorithms 66 (2025).","ista":"Alon Y, Anastos M. 2025. The completion numbers of hamiltonicity and pancyclicity in random graphs. Random Structures and Algorithms. 66(2), e21286.","chicago":"Alon, Yahav, and Michael Anastos. “The Completion Numbers of Hamiltonicity and Pancyclicity in Random Graphs.” <i>Random Structures and Algorithms</i>. Wiley, 2025. <a href=\"https://doi.org/10.1002/rsa.21286\">https://doi.org/10.1002/rsa.21286</a>.","mla":"Alon, Yahav, and Michael Anastos. “The Completion Numbers of Hamiltonicity and Pancyclicity in Random Graphs.” <i>Random Structures and Algorithms</i>, vol. 66, no. 2, e21286, Wiley, 2025, doi:<a href=\"https://doi.org/10.1002/rsa.21286\">10.1002/rsa.21286</a>.","ama":"Alon Y, Anastos M. The completion numbers of hamiltonicity and pancyclicity in random graphs. <i>Random Structures and Algorithms</i>. 2025;66(2). doi:<a href=\"https://doi.org/10.1002/rsa.21286\">10.1002/rsa.21286</a>","ieee":"Y. Alon and M. Anastos, “The completion numbers of hamiltonicity and pancyclicity in random graphs,” <i>Random Structures and Algorithms</i>, vol. 66, no. 2. Wiley, 2025."},"ec_funded":1,"oa_version":"Published Version","_id":"19440","issue":"2","intvolume":"        66"}