{"id":"https://openalex.org/W1991809010","doi":"https://doi.org/10.1145/2213977.2214060","title":"The freezing threshold for k-colourings of a random graph","display_name":"The freezing threshold for k-colourings of a random graph","publication_year":2012,"publication_date":"2012-05-19","ids":{"openalex":"https://openalex.org/W1991809010","doi":"https://doi.org/10.1145/2213977.2214060","mag":"1991809010"},"language":"en","primary_location":{"id":"doi:10.1145/2213977.2214060","is_oa":false,"landing_page_url":"https://doi.org/10.1145/2213977.2214060","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Proceedings of the forty-fourth annual ACM symposium on Theory of computing","raw_type":"proceedings-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5072889645","display_name":"Michael Molloy","orcid":"https://orcid.org/0000-0001-5435-1015"},"institutions":[{"id":"https://openalex.org/I185261750","display_name":"University of Toronto","ror":"https://ror.org/03dbr7087","country_code":"CA","type":"education","lineage":["https://openalex.org/I185261750"]}],"countries":["CA"],"is_corresponding":true,"raw_author_name":"Michael Molloy","raw_affiliation_strings":["University of Toronto, Toronto, ON, Canada","University of Toronto, Toronto, On, Canada"],"affiliations":[{"raw_affiliation_string":"University of Toronto, Toronto, ON, Canada","institution_ids":["https://openalex.org/I185261750"]},{"raw_affiliation_string":"University of Toronto, Toronto, On, Canada","institution_ids":["https://openalex.org/I185261750"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5072889645"],"corresponding_institution_ids":["https://openalex.org/I185261750"],"apc_list":null,"apc_paid":null,"fwci":6.5762,"has_fulltext":false,"cited_by_count":52,"citation_normalized_percentile":{"value":0.97105988,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":94,"max":99},"biblio":{"volume":null,"issue":null,"first_page":"921","last_page":"930"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10374","display_name":"Advanced Graph Theory Research","score":0.996999979019165,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10374","display_name":"Advanced Graph Theory Research","score":0.996999979019165,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11329","display_name":"Limits and Structures in Graph Theory","score":0.9891999959945679,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11152","display_name":"Stochastic processes and statistical mechanics","score":0.9886999726295471,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/random-graph","display_name":"Random graph","score":0.6926558017730713},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.6741881370544434},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5664942264556885},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.5092600584030151},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.45501768589019775},{"id":"https://openalex.org/keywords/random-regular-graph","display_name":"Random regular graph","score":0.45336490869522095},{"id":"https://openalex.org/keywords/1-planar-graph","display_name":"1-planar graph","score":0.24900490045547485},{"id":"https://openalex.org/keywords/line-graph","display_name":"Line graph","score":0.16420024633407593}],"concepts":[{"id":"https://openalex.org/C47458327","wikidata":"https://www.wikidata.org/wiki/Q910404","display_name":"Random graph","level":3,"score":0.6926558017730713},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6741881370544434},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5664942264556885},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.5092600584030151},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.45501768589019775},{"id":"https://openalex.org/C4255713","wikidata":"https://www.wikidata.org/wiki/Q7291988","display_name":"Random regular graph","level":5,"score":0.45336490869522095},{"id":"https://openalex.org/C102192266","wikidata":"https://www.wikidata.org/wiki/Q4545823","display_name":"1-planar graph","level":4,"score":0.24900490045547485},{"id":"https://openalex.org/C203776342","wikidata":"https://www.wikidata.org/wiki/Q1378376","display_name":"Line graph","level":3,"score":0.16420024633407593}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1145/2213977.2214060","is_oa":false,"landing_page_url":"https://doi.org/10.1145/2213977.2214060","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Proceedings of the forty-fourth annual ACM symposium on Theory of computing","raw_type":"proceedings-article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.294.9153","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.294.9153","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.cs.toronto.edu/~molloy/webpapers/frozencolstoc2.pdf","raw_type":"text"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320316838","display_name":"Kempe Foundation","ror":null}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":71,"referenced_works":["https://openalex.org/W4983232","https://openalex.org/W113038098","https://openalex.org/W1620410031","https://openalex.org/W1732919352","https://openalex.org/W1782581930","https://openalex.org/W1818081266","https://openalex.org/W1854875838","https://openalex.org/W1965266146","https://openalex.org/W1971493102","https://openalex.org/W1978113428","https://openalex.org/W1980276492","https://openalex.org/W1980731606","https://openalex.org/W1987347588","https://openalex.org/W1988522098","https://openalex.org/W2000315374","https://openalex.org/W2010193278","https://openalex.org/W2017816219","https://openalex.org/W2030099274","https://openalex.org/W2033482390","https://openalex.org/W2033652300","https://openalex.org/W2036312115","https://openalex.org/W2038932251","https://openalex.org/W2042243015","https://openalex.org/W2044881936","https://openalex.org/W2047467760","https://openalex.org/W2063979426","https://openalex.org/W2065663455","https://openalex.org/W2069603099","https://openalex.org/W2083392982","https://openalex.org/W2089146215","https://openalex.org/W2091476183","https://openalex.org/W2104427163","https://openalex.org/W2105546154","https://openalex.org/W2107779545","https://openalex.org/W2109734774","https://openalex.org/W2110012617","https://openalex.org/W2112575355","https://openalex.org/W2115831572","https://openalex.org/W2126095574","https://openalex.org/W2126216439","https://openalex.org/W2145098470","https://openalex.org/W2152026596","https://openalex.org/W2161885176","https://openalex.org/W2168290833","https://openalex.org/W2333732752","https://openalex.org/W2566505556","https://openalex.org/W2570121038","https://openalex.org/W2621271919","https://openalex.org/W2905110430","https://openalex.org/W2950361113","https://openalex.org/W2963295813","https://openalex.org/W2963880447","https://openalex.org/W3098629286","https://openalex.org/W3103834525","https://openalex.org/W3105049882","https://openalex.org/W3105442588","https://openalex.org/W3105479371","https://openalex.org/W3105799754","https://openalex.org/W3140660692","https://openalex.org/W3142570770","https://openalex.org/W3143219376","https://openalex.org/W3144881883","https://openalex.org/W4213277474","https://openalex.org/W4291225869","https://openalex.org/W4387723197","https://openalex.org/W6636690859","https://openalex.org/W6654604169","https://openalex.org/W6672718537","https://openalex.org/W6676858213","https://openalex.org/W6786599368","https://openalex.org/W6891704858"],"related_works":["https://openalex.org/W4391375266","https://openalex.org/W2012853908","https://openalex.org/W2092565550","https://openalex.org/W2170524605","https://openalex.org/W365256087","https://openalex.org/W2071911310","https://openalex.org/W2610630562","https://openalex.org/W2963998228","https://openalex.org/W2030761932","https://openalex.org/W612643567"],"abstract_inverted_index":{"We":[0,14],"rigorously":[1],"determine":[2],"the":[3,52,66,86,95,99,102,105,111,130,138],"exact":[4],"freezing":[5,96],"threshold,":[6],"rkf,":[7,23,70],"for":[8,17],"k-colourings":[9],"of":[10,33,47,54,101,113],"a":[11,30,45,58,114],"random":[12,18,115],"graph.":[13],"prove":[15],"that":[16,29,38,94,109,132],"graphs":[19],"with":[20],"density":[21,67,134],"above":[22],"almost":[24,72],"every":[25,73],"colouring":[26,74],"is":[27,68,98,135],"such":[28],"linear":[31],"number":[32],"variables":[34],"are":[35,124],"frozen,":[36],"meaning":[37],"their":[39],"colours":[40,53],"cannot":[41],"be":[42],"changed":[43],"by":[44],"sequence":[46],"alterations":[48],"whereby":[49],"we":[50],"change":[51],"o(n)":[55,78],"vertices":[56],"at":[57,76],"time,":[59],"always":[60],"obtaining":[61],"another":[62],"proper":[63],"colouring.":[64],"When":[65],"below":[69],"then":[71],"has":[75,91],"most":[77],"frozen":[79],"variables.":[80],"This":[81],"confirms":[82],"hypotheses":[83],"made":[84],"using":[85],"non-rigorous":[87],"cavity":[88],"method.":[89],"It":[90],"been":[92],"hypothesized":[93],"threshold":[97],"cause":[100],"\"algorithmic":[103],"barrier\",":[104],"long":[106],"observed":[107],"phenomenon":[108],"when":[110],"edge-density":[112],"graph":[116],"exceeds":[117],"hf":[118],"k":[119],"ln":[120],"k(1+ok(1)),":[121],"no":[122],"algorithms":[123],"known":[125],"to":[126],"find":[127],"k-colourings,":[128],"despite":[129],"fact":[131],"this":[133],"only":[136],"half":[137],"k-colourability":[139],"threshold.":[140]},"counts_by_year":[{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":3},{"year":2021,"cited_by_count":3},{"year":2020,"cited_by_count":3},{"year":2019,"cited_by_count":2},{"year":2018,"cited_by_count":8},{"year":2017,"cited_by_count":5},{"year":2016,"cited_by_count":7},{"year":2015,"cited_by_count":8},{"year":2014,"cited_by_count":4},{"year":2013,"cited_by_count":7}],"updated_date":"2026-04-04T16:13:02.066488","created_date":"2025-10-10T00:00:00"}