{"id":"https://openalex.org/W4401672058","doi":"https://doi.org/10.1007/s00373-024-02830-4","title":"The Planar Tur\u00e1n Number of $$\\{K_4,C_5\\}$$ and $$\\{K_4,C_6\\}$$","display_name":"The Planar Tur\u00e1n Number of $$\\{K_4,C_5\\}$$ and $$\\{K_4,C_6\\}$$","publication_year":2024,"publication_date":"2024-08-18","ids":{"openalex":"https://openalex.org/W4401672058","doi":"https://doi.org/10.1007/s00373-024-02830-4"},"language":"lv","primary_location":{"id":"doi:10.1007/s00373-024-02830-4","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00373-024-02830-4","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00373-024-02830-4.pdf","source":{"id":"https://openalex.org/S186126824","display_name":"Graphs and Combinatorics","issn_l":"0911-0119","issn":["0911-0119","1435-5914"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Graphs and Combinatorics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s00373-024-02830-4.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5010046232","display_name":"Ervin Gy\u0151ri","orcid":"https://orcid.org/0000-0001-5691-2242"},"institutions":[{"id":"https://openalex.org/I127817813","display_name":"HUN-REN Alfr\u00e9d R\u00e9nyi Institute of Mathematics","ror":"https://ror.org/03vw74f64","country_code":"HU","type":"facility","lineage":["https://openalex.org/I127817813","https://openalex.org/I7597260"]}],"countries":["HU"],"is_corresponding":false,"raw_author_name":"Ervin Gy\u0151ri","raw_affiliation_strings":["lfr\u00e9d R\u00e9nyi Institute of Mathematics, Budapest, Hungary"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"lfr\u00e9d R\u00e9nyi Institute of Mathematics, Budapest, Hungary","institution_ids":["https://openalex.org/I127817813"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5086020490","display_name":"Alan Li","orcid":null},"institutions":[{"id":"https://openalex.org/I177605424","display_name":"Amherst College","ror":"https://ror.org/028vqfs63","country_code":"US","type":"education","lineage":["https://openalex.org/I177605424"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Alan Li","raw_affiliation_strings":["Amherst College, 220 South Pleasant Street, Amherst, MA, 01002, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Amherst College, 220 South Pleasant Street, Amherst, MA, 01002, USA","institution_ids":["https://openalex.org/I177605424"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5112984649","display_name":"Runtian Zhou","orcid":null},"institutions":[{"id":"https://openalex.org/I170897317","display_name":"Duke University","ror":"https://ror.org/00py81415","country_code":"US","type":"education","lineage":["https://openalex.org/I170897317"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Runtian Zhou","raw_affiliation_strings":["Duke University, 2138 Campus Dr, Durham, NC, 27708, USA"],"raw_orcid":"https://orcid.org/0009-0005-7525-3541","affiliations":[{"raw_affiliation_string":"Duke University, 2138 Campus Dr, Durham, NC, 27708, USA","institution_ids":["https://openalex.org/I170897317"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5112984649"],"corresponding_institution_ids":["https://openalex.org/I170897317"],"apc_list":{"value":2190,"currency":"USD","value_usd":2190},"apc_paid":{"value":2190,"currency":"USD","value_usd":2190},"fwci":1.0991,"has_fulltext":true,"cited_by_count":1,"citation_normalized_percentile":{"value":0.72438672,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":91,"max":95},"biblio":{"volume":"40","issue":"5","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":0.9998000264167786,"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":0.9998000264167786,"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":0.9991000294685364,"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/T11476","display_name":"Graph theory and applications","score":0.9911999702453613,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"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/algorithm","display_name":"Algorithm","score":0.5393140316009521},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.21650728583335876}],"concepts":[{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.5393140316009521},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.21650728583335876}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1007/s00373-024-02830-4","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00373-024-02830-4","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00373-024-02830-4.pdf","source":{"id":"https://openalex.org/S186126824","display_name":"Graphs and Combinatorics","issn_l":"0911-0119","issn":["0911-0119","1435-5914"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Graphs and Combinatorics","raw_type":"journal-article"},{"id":"pmh:oai:real.mtak.hu:221237","is_oa":true,"landing_page_url":"https://real.mtak.hu/221237/1/s00373-024-02830-4.pdf","pdf_url":"https://real.mtak.hu/221237/1/s00373-024-02830-4.pdf","source":{"id":"https://openalex.org/S4306400081","display_name":"Repository of the Academy's Library (Library of the Hungarian Academy of Sciences)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210140733","host_organization_name":"Library and Information Centre of the Hungarian Academy of Sciences","host_organization_lineage":["https://openalex.org/I4210140733"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"acceptedVersion","is_accepted":true,"is_published":false,"raw_source_name":null,"raw_type":"info:eu-repo/semantics/article"}],"best_oa_location":{"id":"doi:10.1007/s00373-024-02830-4","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00373-024-02830-4","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00373-024-02830-4.pdf","source":{"id":"https://openalex.org/S186126824","display_name":"Graphs and Combinatorics","issn_l":"0911-0119","issn":["0911-0119","1435-5914"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Graphs and Combinatorics","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G8550484722","display_name":null,"funder_award_id":"132696","funder_id":"https://openalex.org/F4320326762","funder_display_name":"Nemzeti Kutat\u00e1si Fejleszt\u00e9si \u00e9s Innov\u00e1ci\u00f3s Hivatal"}],"funders":[{"id":"https://openalex.org/F4320326762","display_name":"Nemzeti Kutat\u00e1si Fejleszt\u00e9si \u00e9s Innov\u00e1ci\u00f3s Hivatal","ror":null},{"id":"https://openalex.org/F4320336675","display_name":"National Research, Development and Innovation Office","ror":null}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4401672058.pdf","grobid_xml":"https://content.openalex.org/works/W4401672058.grobid-xml"},"referenced_works_count":4,"referenced_works":["https://openalex.org/W2008078243","https://openalex.org/W2107962892","https://openalex.org/W2965928990","https://openalex.org/W4293581800"],"related_works":["https://openalex.org/W2051487156","https://openalex.org/W2073681303","https://openalex.org/W2053286651","https://openalex.org/W2181743346","https://openalex.org/W2187401768","https://openalex.org/W2181413294","https://openalex.org/W2989452537","https://openalex.org/W2052122378","https://openalex.org/W2544423928","https://openalex.org/W2062023542"],"abstract_inverted_index":{"Abstract":[0],"Let":[1],"$$\\mathcal":[2,44,51],"{H}$$":[3,45],"H":[4,23,46,53,56,73],"be":[5],"a":[6,48],"set":[7],"of":[8,30,43,78],"graphs.":[9],"The":[10,76],"planar":[11,35,80],"Tur\u00e1n":[12],"number,":[13],"$$ex_\\mathcal":[14,65,98,110,130,142,160,187],"{P}(n,\\mathcal":[15],"{H})$$":[16],"e":[17,67,100,112,132,144,164,191],"x":[18,68,101,113,133,145,165,192],"P":[19,69,102,114,134,146,166,193],"(":[20,70,103,115,135,147,167,181,194,208],"n":[21,71,104,116,136,148,168,182,195,209],",":[22,25,72,105,117,137,149,169,173,196,200],")":[24,74,108,120,140,152,177,185,204,212],"is":[26],"the":[27,220],"maximum":[28],"number":[29],"edges":[31],"in":[32],"an":[33],"n-vertex":[34],"graph":[36],"which":[37,218],"does":[38],"not":[39],"contain":[40],"any":[41],"member":[42],"as":[47],"subgraph.":[49],"When":[50],"{H}=\\{H\\}$$":[52],"=":[54],"{":[55,170,197],"}":[57,176,203],"has":[58],"only":[59],"one":[60],"element,":[61],"we":[62,157],"usually":[63],"write":[64],"{P}(n,H)$$":[66],"instead.":[75],"study":[77],"extremal":[79],"graphs":[81],"was":[82],"initiated":[83],"by":[84],"Dowden":[85],"(J":[86],"Graph":[87],"Theory":[88],"83(3):213\u2013230,":[89],"2016).":[90],"He":[91],"obtained":[92],"sharp":[93,124,223],"upper":[94,125],"bounds":[95,126,221],"for":[96,129,224],"both":[97],"{P}(n,C_5)$$":[99],"C":[106,138,150,174,201],"5":[107,175],"and":[109,141,186],"{P}(n,K_4)$$":[111],"K":[118,171,198],"4":[119,172,199],".":[121,153,213],"Later":[122],"on,":[123],"were":[127],"proved":[128],"{P}(n,C_6)$$":[131],"6":[139,202],"{P}(n,C_7)$$":[143],"7":[151,180,206],"In":[154],"this":[155],"paper,":[156],"show":[158,219],"that":[159],"{P}(n,\\{K_4,C_5\\})\\le":[161],"{15\\over":[162],"7}(n-2)$$":[163],"\u2264":[178,205],"15":[179],"-":[183,210],"2":[184,211],"{P}(n,\\{K_4,C_6\\})\\le":[188],"{7\\over":[189],"3}(n-2)$$":[190],"3":[207],"We":[214],"also":[215],"give":[216],"constructions":[217],"are":[222],"infinitely":[225],"many":[226],"n.":[227]},"counts_by_year":[{"year":2025,"cited_by_count":1}],"updated_date":"2026-08-27T14:10:00.468798","created_date":"2025-10-10T00:00:00"}