{"id":"https://openalex.org/W2097900174","doi":"https://doi.org/10.1137/050648237","title":"The Operator $\\Psi$ for the Chromatic Number of a Graph","display_name":"The Operator $\\Psi$ for the Chromatic Number of a Graph","publication_year":2008,"publication_date":"2008-01-01","ids":{"openalex":"https://openalex.org/W2097900174","doi":"https://doi.org/10.1137/050648237","mag":"2097900174"},"language":"en","primary_location":{"id":"doi:10.1137/050648237","is_oa":false,"landing_page_url":"https://doi.org/10.1137/050648237","pdf_url":null,"source":{"id":"https://openalex.org/S928796702","display_name":"SIAM Journal on Optimization","issn_l":"1052-6234","issn":["1052-6234","1095-7189"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Optimization","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://ir.cwi.nl/pub/11832","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5008319452","display_name":"Neboj\u0161a Gvozdenovi\u0107","orcid":"https://orcid.org/0000-0002-9230-9528"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Neboj\u0161a Gvozdenovi\u0107","raw_affiliation_strings":["N.Gvozdenovic@cwi.nl and M.Laurent@cwi.nl#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"N.Gvozdenovic@cwi.nl and M.Laurent@cwi.nl#TAB#","institution_ids":[]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5047819350","display_name":"Monique Laurent","orcid":"https://orcid.org/0000-0001-8474-2121"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Monique Laurent","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":5.3815,"has_fulltext":false,"cited_by_count":51,"citation_normalized_percentile":{"value":0.95730269,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":99},"biblio":{"volume":"19","issue":"2","first_page":"572","last_page":"591"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.9997000098228455,"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/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.9997000098228455,"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/T10374","display_name":"Advanced Graph Theory Research","score":0.9959999918937683,"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/T12072","display_name":"Machine Learning and Algorithms","score":0.9933000206947327,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"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/combinatorics","display_name":"Combinatorics","score":0.83109450340271},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7939649820327759},{"id":"https://openalex.org/keywords/graph-coloring","display_name":"Graph coloring","score":0.5059031844139099},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.47454699873924255},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4727899730205536},{"id":"https://openalex.org/keywords/beta","display_name":"BETA (programming language)","score":0.44409501552581787}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.83109450340271},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7939649820327759},{"id":"https://openalex.org/C76946457","wikidata":"https://www.wikidata.org/wiki/Q504843","display_name":"Graph coloring","level":3,"score":0.5059031844139099},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.47454699873924255},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4727899730205536},{"id":"https://openalex.org/C2776174256","wikidata":"https://www.wikidata.org/wiki/Q830842","display_name":"BETA (programming language)","level":2,"score":0.44409501552581787},{"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/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1137/050648237","is_oa":false,"landing_page_url":"https://doi.org/10.1137/050648237","pdf_url":null,"source":{"id":"https://openalex.org/S928796702","display_name":"SIAM Journal on Optimization","issn_l":"1052-6234","issn":["1052-6234","1095-7189"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Optimization","raw_type":"journal-article"},{"id":"pmh:cwi:oai:cwi.nl:11832","is_oa":true,"landing_page_url":"https://ir.cwi.nl/pub/11832","pdf_url":null,"source":{"id":"https://openalex.org/S4306401843","display_name":"Data Archiving and Networked Services (DANS)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1322597698","host_organization_name":"Royal Netherlands Academy of Arts and Sciences","host_organization_lineage":["https://openalex.org/I1322597698"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"info:eu-repo/semantics/other"},{"id":"pmh:cwi:oai:cwi.nl:12808","is_oa":true,"landing_page_url":"https://ir.cwi.nl/pub/12808","pdf_url":null,"source":{"id":"https://openalex.org/S4306401843","display_name":"Data Archiving and Networked Services (DANS)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1322597698","host_organization_name":"Royal Netherlands Academy of Arts and Sciences","host_organization_lineage":["https://openalex.org/I1322597698"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"SIAM Journal on Optimization, 19(02), 572 - 591","raw_type":"info:eu-repo/semantics/article"},{"id":"pmh:oai:RePEc:ner:tilbur:urn:nbn:nl:ui:12-3959917","is_oa":false,"landing_page_url":"http://arno.uvt.nl/show.cgi?fid=106080","pdf_url":null,"source":{"id":"https://openalex.org/S4306401271","display_name":"RePEc: Research Papers in Economics","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I77793887","host_organization_name":"Federal Reserve Bank of St. Louis","host_organization_lineage":["https://openalex.org/I77793887"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"preprint"}],"best_oa_location":{"id":"pmh:cwi:oai:cwi.nl:11832","is_oa":true,"landing_page_url":"https://ir.cwi.nl/pub/11832","pdf_url":null,"source":{"id":"https://openalex.org/S4306401843","display_name":"Data Archiving and Networked Services (DANS)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1322597698","host_organization_name":"Royal Netherlands Academy of Arts and Sciences","host_organization_lineage":["https://openalex.org/I1322597698"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"info:eu-repo/semantics/other"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":27,"referenced_works":["https://openalex.org/W1526979981","https://openalex.org/W1549791526","https://openalex.org/W1979898269","https://openalex.org/W1998838742","https://openalex.org/W2008495201","https://openalex.org/W2027470314","https://openalex.org/W2043473394","https://openalex.org/W2049143039","https://openalex.org/W2052435807","https://openalex.org/W2056289686","https://openalex.org/W2059453880","https://openalex.org/W2061819450","https://openalex.org/W2069816168","https://openalex.org/W2077397558","https://openalex.org/W2078289947","https://openalex.org/W2082312908","https://openalex.org/W2094911282","https://openalex.org/W2099554026","https://openalex.org/W2106264302","https://openalex.org/W2114189119","https://openalex.org/W2120146316","https://openalex.org/W2121547771","https://openalex.org/W2125240990","https://openalex.org/W2130738665","https://openalex.org/W2145243417","https://openalex.org/W2162579105","https://openalex.org/W4246855957"],"related_works":["https://openalex.org/W2980963558","https://openalex.org/W2606891670","https://openalex.org/W1882246857","https://openalex.org/W3103304230","https://openalex.org/W2185360082","https://openalex.org/W1992365905","https://openalex.org/W4287664878","https://openalex.org/W3212023493","https://openalex.org/W4288278856","https://openalex.org/W2157169356"],"abstract_inverted_index":{"We":[0,15,117],"investigate":[1],"hierarchies":[2],"of":[3,11],"semidefinite":[4],"approximations":[5],"for":[6,85,96,115,126],"the":[7,27,68,82,129,159],"chromatic":[8,70],"number":[9,29,71,133],"$\\chi(G)$":[10,113],"a":[12,35],"graph":[13,22,37,64],"G.":[14],"introduce":[16],"an":[17,56],"operator":[18],"$\\Psi$":[19],"mapping":[20],"any":[21],"parameter":[23,38,65],"$\\beta(G)$,":[24],"nested":[25,40,66],"between":[26,41,67],"stability":[28],"$\\alpha(G)$":[30],"and":[31,45,73,92,141,151,164],"$\\chi(\\overline":[32],"G)$,":[33],"to":[34],"new":[36,120],"$\\Psi_\\beta(G)$,":[39],"$\\alpha":[42],"(\\overline":[43],"G)$":[44],"$\\chi(G)$;":[46],"$\\Psi_\\beta(G)$":[47,110],"is":[48,59],"polynomial":[49,61,121],"time":[50,62,122],"computable":[51,63,123],"if":[52],"$\\beta(G)$":[53],"is.":[54],"As":[55],"application,":[57],"there":[58],"no":[60],"fractional":[69],"$\\chi^*(\\cdot)$":[72],"$\\chi(\\cdot)$":[74],"unless":[75],"$\\text{P}":[76],"=":[77],"\\text{NP}$.":[78],"Moreover,":[79],"based":[80],"on":[81,146],"Motzkin\u2013Straus":[83],"formulation":[84],"$\\alpha(G)$,":[86],"we":[87],"give":[88],"(quadratically":[89],"constrained)":[90],"quadratic":[91],"copositive":[93],"programming":[94],"formulations":[95],"$\\chi(G)$.":[97],"Under":[98],"some":[99],"mild":[100],"assumptions,":[101],"$n/\\beta(G)\\le":[102],"\\Psi_\\beta(G)$,":[103],"but,":[104],"while":[105],"$n/\\beta(G)$":[106],"remains":[107],"below":[108],"$\\chi^*(G)$,":[109],"can":[111],"reach":[112],"(e.g.,":[114],"$\\beta(\\cdot)=\\alpha(\\cdot)$).":[116],"also":[118],"define":[119],"lower":[124],"bounds":[125],"$\\chi(G)$,":[127],"improving":[128],"classic":[130],"Lov\u00e1sz":[131],"theta":[132],"(and":[134],"its":[135],"strengthenings":[136],"obtained":[137],"by":[138],"adding":[139],"nonnegativity":[140],"triangle":[142],"inequalities);":[143],"experimental":[144],"results":[145],"Hamming":[147],"graphs,":[148,150],"Kneser":[149],"DIMACS":[152],"benchmark":[153],"graphs":[154],"will":[155],"be":[156],"given":[157],"in":[158],"follow-up":[160],"paper":[161],"[N.":[162],"Gvozdenovi\u0107":[163],"M.":[165],"Laurent,":[166],"SIAM":[167],"J.":[168],"Optim.,":[169],"19":[170],"(2008),":[171],"pp.":[172],"592\u2013615].":[173]},"counts_by_year":[{"year":2025,"cited_by_count":3},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":5},{"year":2020,"cited_by_count":2},{"year":2019,"cited_by_count":1},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":3},{"year":2016,"cited_by_count":3},{"year":2015,"cited_by_count":6},{"year":2014,"cited_by_count":4},{"year":2013,"cited_by_count":8}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}