{"id":"https://openalex.org/W1882246857","doi":"https://doi.org/10.37236/272","title":"Bipartite Coverings and the Chromatic Number","display_name":"Bipartite Coverings and the Chromatic Number","publication_year":2009,"publication_date":"2009-11-30","ids":{"openalex":"https://openalex.org/W1882246857","doi":"https://doi.org/10.37236/272","mag":"1882246857"},"language":"en","primary_location":{"id":"doi:10.37236/272","is_oa":true,"landing_page_url":"https://doi.org/10.37236/272","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v16i1n34/pdf","source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v16i1n34/pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5079803922","display_name":"Dhruv Mubayi","orcid":"https://orcid.org/0000-0002-4709-8768"},"institutions":[{"id":"https://openalex.org/I39422238","display_name":"University of Illinois Chicago","ror":"https://ror.org/02mpq6x41","country_code":"US","type":"education","lineage":["https://openalex.org/I39422238"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Dhruv Mubayi","raw_affiliation_strings":["Department of Mathematics Statistics, and Computer Science University of Illinois Chicago, IL 60607, USA","University of Illinois at Chicago"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics Statistics, and Computer Science University of Illinois Chicago, IL 60607, USA","institution_ids":["https://openalex.org/I39422238"]},{"raw_affiliation_string":"University of Illinois at Chicago","institution_ids":["https://openalex.org/I39422238"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5053966865","display_name":"Sundar Vishwanathan","orcid":null},"institutions":[{"id":"https://openalex.org/I162827531","display_name":"Indian Institute of Technology Bombay","ror":"https://ror.org/02qyf5152","country_code":"IN","type":"education","lineage":["https://openalex.org/I162827531"]}],"countries":["IN"],"is_corresponding":false,"raw_author_name":"Sundar Vishwanathan","raw_affiliation_strings":["Department of Computer Science Indian Institute of Technology Mumbai India 400076"],"affiliations":[{"raw_affiliation_string":"Department of Computer Science Indian Institute of Technology Mumbai India 400076","institution_ids":["https://openalex.org/I162827531"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5079803922"],"corresponding_institution_ids":["https://openalex.org/I39422238"],"apc_list":null,"apc_paid":null,"fwci":0.4734,"has_fulltext":true,"cited_by_count":6,"citation_normalized_percentile":{"value":0.53455425,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":94},"biblio":{"volume":"16","issue":"1","first_page":null,"last_page":null},"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":0.9991000294685364,"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":0.9991000294685364,"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":0.944599986076355,"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/combinatorics","display_name":"Combinatorics","score":0.9380637407302856},{"id":"https://openalex.org/keywords/bipartite-graph","display_name":"Bipartite graph","score":0.8638427257537842},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8522721529006958},{"id":"https://openalex.org/keywords/conjecture","display_name":"Conjecture","score":0.6923688054084778},{"id":"https://openalex.org/keywords/partition","display_name":"Partition (number theory)","score":0.6362926363945007},{"id":"https://openalex.org/keywords/complete-bipartite-graph","display_name":"Complete bipartite graph","score":0.6341858506202698},{"id":"https://openalex.org/keywords/clique","display_name":"Clique","score":0.5493296980857849},{"id":"https://openalex.org/keywords/chromatic-scale","display_name":"Chromatic scale","score":0.4685947000980377},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.4611611068248749},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.4424195885658264},{"id":"https://openalex.org/keywords/cover","display_name":"Cover (algebra)","score":0.4322543740272522},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4208522439002991}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.9380637407302856},{"id":"https://openalex.org/C197657726","wikidata":"https://www.wikidata.org/wiki/Q174733","display_name":"Bipartite graph","level":3,"score":0.8638427257537842},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8522721529006958},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.6923688054084778},{"id":"https://openalex.org/C42812","wikidata":"https://www.wikidata.org/wiki/Q1082910","display_name":"Partition (number theory)","level":2,"score":0.6362926363945007},{"id":"https://openalex.org/C134119311","wikidata":"https://www.wikidata.org/wiki/Q913598","display_name":"Complete bipartite graph","level":4,"score":0.6341858506202698},{"id":"https://openalex.org/C2777035058","wikidata":"https://www.wikidata.org/wiki/Q1662634","display_name":"Clique","level":2,"score":0.5493296980857849},{"id":"https://openalex.org/C196956537","wikidata":"https://www.wikidata.org/wiki/Q202021","display_name":"Chromatic scale","level":2,"score":0.4685947000980377},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.4611611068248749},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.4424195885658264},{"id":"https://openalex.org/C2780428219","wikidata":"https://www.wikidata.org/wiki/Q16952335","display_name":"Cover (algebra)","level":2,"score":0.4322543740272522},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4208522439002991},{"id":"https://openalex.org/C78519656","wikidata":"https://www.wikidata.org/wiki/Q101333","display_name":"Mechanical engineering","level":1,"score":0.0},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.0},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.37236/272","is_oa":true,"landing_page_url":"https://doi.org/10.37236/272","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v16i1n34/pdf","source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.156.6662","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.156.6662","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.math.uic.edu/~mubayi/papers/biclique.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.214.9","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.214.9","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.combinatorics.org/Volume_16/PDF/v16i1n34.pdf","raw_type":"text"},{"id":"pmh:oai:dsapce.library.iitb.ac.in:10054/5904","is_oa":false,"landing_page_url":"http://dspace.library.iitb.ac.in/xmlui/handle/10054/5904","pdf_url":null,"source":{"id":"https://openalex.org/S4306400899","display_name":"DSpace (IIT Bombay)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I162827531","host_organization_name":"Indian Institute of Technology Bombay","host_organization_lineage":["https://openalex.org/I162827531"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Article"}],"best_oa_location":{"id":"doi:10.37236/272","is_oa":true,"landing_page_url":"https://doi.org/10.37236/272","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v16i1n34/pdf","source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G7027789630","display_name":"Problems in Extremal Set Theory","funder_award_id":"0653946","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W1882246857.pdf","grobid_xml":"https://content.openalex.org/works/W1882246857.grobid-xml"},"referenced_works_count":5,"referenced_works":["https://openalex.org/W1984375054","https://openalex.org/W2068380082","https://openalex.org/W2785506964","https://openalex.org/W4285719527","https://openalex.org/W6600876140"],"related_works":["https://openalex.org/W2091826982","https://openalex.org/W2391237956","https://openalex.org/W2036057584","https://openalex.org/W245205139","https://openalex.org/W2921162226","https://openalex.org/W4234658180","https://openalex.org/W2107875203","https://openalex.org/W4235849992","https://openalex.org/W4246000090","https://openalex.org/W1584510240"],"abstract_inverted_index":{"Consider":[0],"a":[1,9,35,98,115],"graph":[2],"$G$":[3,113],"with":[4],"chromatic":[5],"number":[6,44],"$k$":[7],"and":[8,102],"collection":[10],"of":[11,21,37,40,55,60,78,81,100],"complete":[12],"bipartite":[13,32,83],"graphs,":[14],"or":[15],"bicliques,":[16],"that":[17,68,106],"cover":[18,87],"the":[19,25,31,38,52,56,64,79,82,86,107],"edges":[20,39],"$G$.":[22],"We":[23],"prove":[24],"following":[26],"two":[27],"results:":[28],"$\\bullet$":[29,75],"If":[30],"graphs":[33,84],"form":[34],"partition":[36],"$G$,":[41],"then":[42],"their":[43],"is":[45,51,88,109,114],"at":[46,89],"least":[47,90],"$2^{\\sqrt{\\log_2":[48],"k}}$.":[49],"This":[50,93],"first":[53],"improvement":[54],"easy":[57],"lower":[58],"bound":[59],"$\\log_2":[61],"k$,":[62],"while":[63],"Alon-Saks-Seymour":[65],"conjecture":[66],"states":[67],"this":[69],"can":[70],"be":[71],"improved":[72],"to":[73],"$k-1$.":[74],"The":[76],"sum":[77],"orders":[80],"in":[85,95],"$(1-o(1))k\\log_2":[91],"k$.":[92],"generalizes,":[94],"asymptotic":[96],"form,":[97],"result":[99],"Katona":[101],"Szemer\u00e9di":[103],"who":[104],"proved":[105],"minimum":[108],"$k\\log_2":[110],"k$":[111],"when":[112],"clique.":[116]},"counts_by_year":[{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":1},{"year":2020,"cited_by_count":1},{"year":2016,"cited_by_count":1},{"year":2014,"cited_by_count":1},{"year":2012,"cited_by_count":1}],"updated_date":"2026-04-10T15:06:20.359241","created_date":"2025-10-10T00:00:00"}