{"id":"https://openalex.org/W2030362910","doi":"https://doi.org/10.1017/s0963548309990216","title":"The <i>t</i>-Improper Chromatic Number of Random Graphs","display_name":"The <i>t</i>-Improper Chromatic Number of Random Graphs","publication_year":2009,"publication_date":"2009-09-09","ids":{"openalex":"https://openalex.org/W2030362910","doi":"https://doi.org/10.1017/s0963548309990216","mag":"2030362910"},"language":"en","primary_location":{"id":"doi:10.1017/s0963548309990216","is_oa":false,"landing_page_url":"https://doi.org/10.1017/s0963548309990216","pdf_url":null,"source":{"id":"https://openalex.org/S7291079","display_name":"Combinatorics Probability Computing","issn_l":"0963-5483","issn":["0963-5483","1469-2163"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310311721","host_organization_name":"Cambridge University Press","host_organization_lineage":["https://openalex.org/P4310311721","https://openalex.org/P4310311702"],"host_organization_lineage_names":["Cambridge University Press","University of Cambridge"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Combinatorics, Probability and Computing","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/0809.4726","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5019911064","display_name":"Ross J. Kang","orcid":"https://orcid.org/0000-0002-4219-593X"},"institutions":[{"id":"https://openalex.org/I5023651","display_name":"McGill University","ror":"https://ror.org/01pxwe438","country_code":"CA","type":"education","lineage":["https://openalex.org/I5023651"]}],"countries":["CA"],"is_corresponding":true,"raw_author_name":"ROSS J. KANG","raw_affiliation_strings":["School of Computer Science, McGill University, Montr\u00e9al, Qu\u00e9bec, H2A 2A7, Canada (e-mail:","School of computer science, mcgill university, montr\u00e9al, qu\u00e9bec, h2a 2a7, canada (e-mail: rosskang@cs.mcgill.ca)#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Computer Science, McGill University, Montr\u00e9al, Qu\u00e9bec, H2A 2A7, Canada (e-mail:","institution_ids":["https://openalex.org/I5023651"]},{"raw_affiliation_string":"School of computer science, mcgill university, montr\u00e9al, qu\u00e9bec, h2a 2a7, canada (e-mail: rosskang@cs.mcgill.ca)#TAB#","institution_ids":["https://openalex.org/I5023651"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5111437501","display_name":"Colin McDiarmid","orcid":null},"institutions":[{"id":"https://openalex.org/I40120149","display_name":"University of Oxford","ror":"https://ror.org/052gg0110","country_code":"GB","type":"education","lineage":["https://openalex.org/I40120149"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"COLIN McDIARMID","raw_affiliation_strings":["Department of Statistics, University of Oxford, 1 South Parks Road, Oxford OX1 3TG, UK (e-mail:","Department of Statistics, University of Oxford, 1 South Parks Road, Oxford OX1 3TG, UK (e-mail: cmcd@stats.ox.ac.uk)#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Statistics, University of Oxford, 1 South Parks Road, Oxford OX1 3TG, UK (e-mail:","institution_ids":["https://openalex.org/I40120149"]},{"raw_affiliation_string":"Department of Statistics, University of Oxford, 1 South Parks Road, Oxford OX1 3TG, UK (e-mail: cmcd@stats.ox.ac.uk)#TAB#","institution_ids":["https://openalex.org/I40120149"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5019911064"],"corresponding_institution_ids":["https://openalex.org/I5023651"],"apc_list":null,"apc_paid":null,"fwci":3.3466,"has_fulltext":false,"cited_by_count":26,"citation_normalized_percentile":{"value":0.91683778,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":97},"biblio":{"volume":"19","issue":"1","first_page":"87","last_page":"98"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11329","display_name":"Limits and Structures in Graph Theory","score":0.9911999702453613,"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"}},"topics":[{"id":"https://openalex.org/T11329","display_name":"Limits and Structures in Graph Theory","score":0.9911999702453613,"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/T10374","display_name":"Advanced Graph Theory Research","score":0.9495999813079834,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/chromatic-scale","display_name":"Chromatic scale","score":0.7984375357627869},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.7946359515190125},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7303462028503418},{"id":"https://openalex.org/keywords/random-graph","display_name":"Random graph","score":0.6329749822616577},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.5482476353645325},{"id":"https://openalex.org/keywords/constant","display_name":"Constant (computer programming)","score":0.4598560929298401},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4346208870410919},{"id":"https://openalex.org/keywords/degree","display_name":"Degree (music)","score":0.42304784059524536},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.13913363218307495},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.09805566072463989}],"concepts":[{"id":"https://openalex.org/C196956537","wikidata":"https://www.wikidata.org/wiki/Q202021","display_name":"Chromatic scale","level":2,"score":0.7984375357627869},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7946359515190125},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7303462028503418},{"id":"https://openalex.org/C47458327","wikidata":"https://www.wikidata.org/wiki/Q910404","display_name":"Random graph","level":3,"score":0.6329749822616577},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.5482476353645325},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.4598560929298401},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4346208870410919},{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.42304784059524536},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.13913363218307495},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.09805566072463989},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C24890656","wikidata":"https://www.wikidata.org/wiki/Q82811","display_name":"Acoustics","level":1,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1017/s0963548309990216","is_oa":false,"landing_page_url":"https://doi.org/10.1017/s0963548309990216","pdf_url":null,"source":{"id":"https://openalex.org/S7291079","display_name":"Combinatorics Probability Computing","issn_l":"0963-5483","issn":["0963-5483","1469-2163"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310311721","host_organization_name":"Cambridge University Press","host_organization_lineage":["https://openalex.org/P4310311721","https://openalex.org/P4310311702"],"host_organization_lineage_names":["Cambridge University Press","University of Cambridge"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Combinatorics, Probability and Computing","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:0809.4726","is_oa":true,"landing_page_url":"http://arxiv.org/abs/0809.4726","pdf_url":"https://arxiv.org/pdf/0809.4726","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.311.2941","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.311.2941","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://arxiv.org/pdf/0809.4726v2.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.311.8319","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.311.8319","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://arxiv.org/pdf/0809.4726v1.pdf","raw_type":"text"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:0809.4726","is_oa":true,"landing_page_url":"http://arxiv.org/abs/0809.4726","pdf_url":"https://arxiv.org/pdf/0809.4726","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":29,"referenced_works":["https://openalex.org/W264180933","https://openalex.org/W390146837","https://openalex.org/W623244084","https://openalex.org/W1927650282","https://openalex.org/W1987487523","https://openalex.org/W2009827511","https://openalex.org/W2012724629","https://openalex.org/W2018242275","https://openalex.org/W2018650020","https://openalex.org/W2032265004","https://openalex.org/W2059120410","https://openalex.org/W2082249316","https://openalex.org/W2103001299","https://openalex.org/W2105645406","https://openalex.org/W2112575355","https://openalex.org/W2120503287","https://openalex.org/W2129414351","https://openalex.org/W2145878357","https://openalex.org/W2165341195","https://openalex.org/W2478722229","https://openalex.org/W2495758475","https://openalex.org/W2569553364","https://openalex.org/W2905110430","https://openalex.org/W2953123155","https://openalex.org/W3144881883","https://openalex.org/W4241684091","https://openalex.org/W4285719527","https://openalex.org/W4301258660","https://openalex.org/W6609756778"],"related_works":["https://openalex.org/W2399035783","https://openalex.org/W3161230432","https://openalex.org/W4367190712","https://openalex.org/W2168900540","https://openalex.org/W1487201914","https://openalex.org/W4301429470","https://openalex.org/W280458463","https://openalex.org/W1170131723","https://openalex.org/W2125310826","https://openalex.org/W3178681420"],"abstract_inverted_index":{"We":[0],"consider":[1],"the":[2,8,26,36,60,67,79,90,95],"t":[3,16,21,51,54,84,98,100],"-improper":[4,17],"chromatic":[5,18],"number":[6,19,28],"of":[7,29,35,46,63,78,82,92,97],"Erd\u0151s\u2013R\u00e9nyi":[9],"random":[10],"graph":[11],"G":[12,23,86],"n,p":[13,87],".":[14,52],"The":[15],"\u03c7":[20,83],"(":[22,85,101],")":[24,88],"is":[25,59,71],"smallest":[27],"colours":[30],"needed":[31],"in":[32,38],"a":[33,44,75],"colouring":[34],"vertices":[37],"which":[39],"each":[40],"colour":[41],"class":[42],"induces":[43],"subgraph":[45],"maximum":[47],"degree":[48],"at":[49],"most":[50],"If":[53],"=":[55,99],"0,":[56],"then":[57],"this":[58],"usual":[61],"notion":[62],"proper":[64],"colouring.":[65],"When":[66],"edge":[68],"probability":[69],"p":[70],"constant,":[72],"we":[73],"provide":[74],"detailed":[76],"description":[77],"asymptotic":[80],"behaviour":[81],"over":[89],"range":[91],"choices":[93],"for":[94],"growth":[96],"n":[102],").":[103]},"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2022,"cited_by_count":1},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":2},{"year":2018,"cited_by_count":2},{"year":2017,"cited_by_count":1},{"year":2016,"cited_by_count":2},{"year":2015,"cited_by_count":3},{"year":2014,"cited_by_count":2},{"year":2013,"cited_by_count":1},{"year":2012,"cited_by_count":2}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2016-06-24T00:00:00"}