{"id":"https://openalex.org/W2513044153","doi":"https://doi.org/10.37236/5309","title":"Planar Graphs have Independence Ratio at least 3/13","display_name":"Planar Graphs have Independence Ratio at least 3/13","publication_year":2016,"publication_date":"2016-09-02","ids":{"openalex":"https://openalex.org/W2513044153","doi":"https://doi.org/10.37236/5309","mag":"2513044153"},"language":"en","primary_location":{"id":"doi:10.37236/5309","is_oa":true,"landing_page_url":"https://doi.org/10.37236/5309","pdf_url":null,"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},"type":"preprint","indexed_in":["arxiv","crossref","datacite","doaj"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://doi.org/10.37236/5309","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5064235428","display_name":"Daniel W. Cranston","orcid":"https://orcid.org/0000-0003-3592-6105"},"institutions":[{"id":"https://openalex.org/I184840846","display_name":"Virginia Commonwealth University","ror":"https://ror.org/02nkdxk79","country_code":"US","type":"education","lineage":["https://openalex.org/I184840846"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Daniel W. Cranston","raw_affiliation_strings":["Virginia Commonwealth University"],"affiliations":[{"raw_affiliation_string":"Virginia Commonwealth University","institution_ids":["https://openalex.org/I184840846"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5017927005","display_name":"Landon Rabern","orcid":"https://orcid.org/0000-0002-8075-6806"},"institutions":[{"id":"https://openalex.org/I173146385","display_name":"Franklin & Marshall College","ror":"https://ror.org/04fp4ps48","country_code":"US","type":"education","lineage":["https://openalex.org/I173146385"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Landon Rabern","raw_affiliation_strings":["Franklin & Marshall"],"affiliations":[{"raw_affiliation_string":"Franklin & Marshall","institution_ids":["https://openalex.org/I173146385"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5064235428"],"corresponding_institution_ids":["https://openalex.org/I184840846"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.07316479,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"23","issue":"3","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11522","display_name":"VLSI and FPGA Design Techniques","score":0.9988999962806702,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11522","display_name":"VLSI and FPGA Design Techniques","score":0.9988999962806702,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10996","display_name":"Computational Geometry and Mesh Generation","score":0.9975000023841858,"subfield":{"id":"https://openalex.org/subfields/1704","display_name":"Computer Graphics and Computer-Aided Design"},"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":0.9962000250816345,"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/independence-number","display_name":"Independence number","score":0.8221257925033569},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.7764867544174194},{"id":"https://openalex.org/keywords/planar-graph","display_name":"Planar graph","score":0.6585060954093933},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6566868424415588},{"id":"https://openalex.org/keywords/planar","display_name":"Planar","score":0.5930781960487366},{"id":"https://openalex.org/keywords/disjoint-sets","display_name":"Disjoint sets","score":0.5754144191741943},{"id":"https://openalex.org/keywords/vertex","display_name":"Vertex (graph theory)","score":0.5436269044876099},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.538066029548645},{"id":"https://openalex.org/keywords/independent-set","display_name":"Independent set","score":0.474084734916687},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.47340235114097595},{"id":"https://openalex.org/keywords/independence","display_name":"Independence (probability theory)","score":0.45706358551979065},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.40798673033714294},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.10286638140678406},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.09866306185722351},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.05105474591255188}],"concepts":[{"id":"https://openalex.org/C2984853995","wikidata":"https://www.wikidata.org/wiki/Q1060343","display_name":"Independence number","level":3,"score":0.8221257925033569},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7764867544174194},{"id":"https://openalex.org/C101837359","wikidata":"https://www.wikidata.org/wiki/Q547823","display_name":"Planar graph","level":3,"score":0.6585060954093933},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6566868424415588},{"id":"https://openalex.org/C134786449","wikidata":"https://www.wikidata.org/wiki/Q3391255","display_name":"Planar","level":2,"score":0.5930781960487366},{"id":"https://openalex.org/C45340560","wikidata":"https://www.wikidata.org/wiki/Q215382","display_name":"Disjoint sets","level":2,"score":0.5754144191741943},{"id":"https://openalex.org/C80899671","wikidata":"https://www.wikidata.org/wiki/Q1304193","display_name":"Vertex (graph theory)","level":3,"score":0.5436269044876099},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.538066029548645},{"id":"https://openalex.org/C122818955","wikidata":"https://www.wikidata.org/wiki/Q1060343","display_name":"Independent set","level":3,"score":0.474084734916687},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.47340235114097595},{"id":"https://openalex.org/C35651441","wikidata":"https://www.wikidata.org/wiki/Q625303","display_name":"Independence (probability theory)","level":2,"score":0.45706358551979065},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.40798673033714294},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.10286638140678406},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.09866306185722351},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.05105474591255188},{"id":"https://openalex.org/C121684516","wikidata":"https://www.wikidata.org/wiki/Q7600677","display_name":"Computer graphics (images)","level":1,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.37236/5309","is_oa":true,"landing_page_url":"https://doi.org/10.37236/5309","pdf_url":null,"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1609.06010","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1609.06010","pdf_url":"https://arxiv.org/pdf/1609.06010","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":"","raw_type":"text"},{"id":"mag:2513044153","is_oa":true,"landing_page_url":"https://arxiv.org/pdf/1609.06010.pdf","pdf_url":null,"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":"arXiv (Cornell University)","raw_type":null},{"id":"doi:10.48550/arxiv.1609.06010","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.1609.06010","pdf_url":null,"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":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article-journal"}],"best_oa_location":{"id":"doi:10.37236/5309","is_oa":true,"landing_page_url":"https://doi.org/10.37236/5309","pdf_url":null,"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":3,"referenced_works":["https://openalex.org/W1970462054","https://openalex.org/W2020112047","https://openalex.org/W2148360835"],"related_works":["https://openalex.org/W3047592838","https://openalex.org/W1650458931","https://openalex.org/W3123442066","https://openalex.org/W1792309686","https://openalex.org/W2950813788","https://openalex.org/W3178441973","https://openalex.org/W2964248410","https://openalex.org/W2209821098","https://openalex.org/W3115851297","https://openalex.org/W3042354966","https://openalex.org/W3005027787","https://openalex.org/W2041857266","https://openalex.org/W2949626445","https://openalex.org/W2281501900","https://openalex.org/W2949795080","https://openalex.org/W2338527069","https://openalex.org/W2002377720","https://openalex.org/W2949884747","https://openalex.org/W3016091878","https://openalex.org/W3138467413"],"abstract_inverted_index":{"The":[0],"4":[1],"Color":[2],"Theorem":[3],"(4CT)":[4],"implies":[5],"that":[6,62],"every":[7,63],"$n$-vertex":[8,64],"planar":[9,65],"graph":[10,66],"has":[11,67],"an":[12,68],"independent":[13,69,83],"set":[14,70],"of":[15,30,33,59,71,84],"size":[16,72],"at":[17,73],"least":[18,74],"$\\frac{n}4$;":[19],"this":[20,40,78,91],"is":[21],"best":[22,81],"possible,":[23],"as":[24],"shown":[25],"by":[26],"the":[27,51,60,80,85],"disjoint":[28],"union":[29],"many":[31],"copies":[32],"$K_4$.":[34],"In":[35,54],"1968,":[36],"Erd\u0151s":[37],"asked":[38],"whether":[39],"bound":[41,82,92],"on":[42],"independence":[43],"number":[44],"could":[45],"be":[46],"proved":[47],"more":[48],"easily":[49],"than":[50],"full":[52],"4CT.":[53,86],"1976":[55],"Albertson":[56],"showed":[57],"(independently":[58],"4CT)":[61],"$\\frac{2n}9$.":[75],"Until":[76],"now,":[77],"remained":[79],"Our":[87],"main":[88],"result":[89],"improves":[90],"to":[93],"$\\frac{3n}{13}$.":[94]},"counts_by_year":[],"updated_date":"2026-02-09T09:26:11.010843","created_date":"2025-10-10T00:00:00"}