{"id":"https://openalex.org/W7214711568","doi":"https://doi.org/10.48550/arxiv.2609.33902","title":"Det-extremal cubic graphs and the total domatic number","display_name":"Det-extremal cubic graphs and the total domatic number","publication_year":2026,"publication_date":"2026-09-27","ids":{"openalex":"https://openalex.org/W7214711568","doi":"https://doi.org/10.48550/arxiv.2609.33902"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.33902","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.33902","pdf_url":"https://arxiv.org/pdf/2609.33902","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2609.33902","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5056963249","display_name":"Myungho Choi","orcid":"https://orcid.org/0000-0001-8921-1565","observed_orcids":["https://orcid.org/0000-0001-8921-1565"]},"institutions":[{"id":"https://openalex.org/I139264467","display_name":"Seoul National University","ror":"https://ror.org/04h9pn542","country_code":"KR","type":"education","lineage":["https://openalex.org/I139264467"]}],"countries":["KR"],"is_corresponding":false,"raw_author_name":"Choi, Myungho","raw_affiliation_strings":["Department of Mathematics Education , Seoul National University , Seoul , Republic of Korea"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics Education , Seoul National University , Seoul , Republic of Korea","institution_ids":["https://openalex.org/I139264467"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5027111662","display_name":"Hyemin Kwon","orcid":"https://orcid.org/0009-0009-0123-8831","observed_orcids":["https://orcid.org/0009-0009-0123-8831"]},"institutions":[{"id":"https://openalex.org/I125608817","display_name":"Korea Institute for Advanced Study","ror":"https://ror.org/041hz9568","country_code":"KR","type":"facility","lineage":["https://openalex.org/I125608817"]}],"countries":["KR"],"is_corresponding":false,"raw_author_name":"Kwon, Hyemin","raw_affiliation_strings":["Korea Institute for Advanced Study (KIAS) , Seoul , Republic of Korea"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Korea Institute for Advanced Study (KIAS) , Seoul , Republic of Korea","institution_ids":["https://openalex.org/I125608817"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5153426185","display_name":"Boram Park","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I139264467","display_name":"Seoul National University","ror":"https://ror.org/04h9pn542","country_code":"KR","type":"education","lineage":["https://openalex.org/I139264467"]}],"countries":["KR"],"is_corresponding":false,"raw_author_name":"Park, Boram","raw_affiliation_strings":["Department of Mathematics Education , Seoul National University , Seoul , Republic of Korea"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics Education , Seoul National University , Seoul , Republic of Korea","institution_ids":["https://openalex.org/I139264467"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":2,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10374","display_name":"Advanced Graph Theory Research","score":0.9883000254631042,"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/T10374","display_name":"Advanced Graph Theory Research","score":0.9883000254631042,"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":0.007000000216066837,"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/T11329","display_name":"Limits and Structures in Graph Theory","score":0.001500000013038516,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/cubic-graphs","display_name":"cubic graphs","score":0.8690000176429749},{"id":"https://openalex.org/keywords/heawood-graph","display_name":"Heawood graph","score":0.9169999957084656},{"id":"https://openalex.org/keywords/3-connected-graphs","display_name":"3-connected graphs","score":0.8360000252723694},{"id":"https://openalex.org/keywords/girth","display_name":"girth","score":0.9039999842643738},{"id":"https://openalex.org/keywords/blocking-sets","display_name":"blocking sets","score":0.8460000157356262},{"id":"https://openalex.org/keywords/total-dominating-sets","display_name":"total dominating sets","score":0.8899999856948853}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.819599986076355},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7889000177383423},{"id":"https://openalex.org/C100500283","wikidata":"https://www.wikidata.org/wiki/Q1374495","display_name":"Cubic graph","level":5,"score":0.5629000067710876},{"id":"https://openalex.org/C68996132","wikidata":"https://www.wikidata.org/wiki/Q3115485","display_name":"Foster graph","level":5,"score":0.5041999816894531},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4993000030517578},{"id":"https://openalex.org/C91429207","wikidata":"https://www.wikidata.org/wiki/Q5192245","display_name":"Cubic form","level":2,"score":0.4595000147819519},{"id":"https://openalex.org/C45340560","wikidata":"https://www.wikidata.org/wiki/Q215382","display_name":"Disjoint sets","level":2,"score":0.41839998960494995},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.41440001130104065},{"id":"https://openalex.org/C197657726","wikidata":"https://www.wikidata.org/wiki/Q174733","display_name":"Bipartite graph","level":3,"score":0.41029998660087585},{"id":"https://openalex.org/C180356752","wikidata":"https://www.wikidata.org/wiki/Q727035","display_name":"Adjacency matrix","level":3,"score":0.38019999861717224},{"id":"https://openalex.org/C114727528","wikidata":"https://www.wikidata.org/wiki/Q7840044","display_name":"Triangle-free graph","level":5,"score":0.3127000033855438},{"id":"https://openalex.org/C59824394","wikidata":"https://www.wikidata.org/wiki/Q5141281","display_name":"Cograph","level":5,"score":0.28929999470710754},{"id":"https://openalex.org/C193435613","wikidata":"https://www.wikidata.org/wiki/Q2997928","display_name":"Connected component","level":2,"score":0.2766999900341034},{"id":"https://openalex.org/C134119311","wikidata":"https://www.wikidata.org/wiki/Q913598","display_name":"Complete bipartite graph","level":4,"score":0.2759000062942505}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.33902","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.33902","pdf_url":"https://arxiv.org/pdf/2609.33902","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":"doi:10.48550/arxiv.2609.33902","doi":"https://doi.org/10.48550/arxiv.2609.33902","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.33902","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.33902","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.33902","pdf_url":"https://arxiv.org/pdf/2609.33902","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":[],"study_designs":[],"awards":[{"id":"https://openalex.org/G6329434893","display_name":null,"funder_award_id":"RS-2025-00523206","funder_id":"https://openalex.org/F4320328359","funder_display_name":"Ministry of Science and ICT, South Korea"},{"id":"https://openalex.org/G7287545","display_name":null,"funder_award_id":"RS-2025-00523206","funder_id":"https://openalex.org/F4320322120","funder_display_name":"National Research Foundation of Korea"}],"funders":[{"id":"https://openalex.org/F4320321292","display_name":"Seoul National University","ror":"https://ror.org/04h9pn542"},{"id":"https://openalex.org/F4320322120","display_name":"National Research Foundation of Korea","ror":"https://ror.org/013aysd81"},{"id":"https://openalex.org/F4320328359","display_name":"Ministry of Science and ICT, South Korea","ror":"https://ror.org/01wpjm123"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7214711568.pdf","grobid_xml":"https://content.openalex.org/works/W7214711568.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W2292154390","https://openalex.org/W139632646","https://openalex.org/W1976691702","https://openalex.org/W2779162822","https://openalex.org/W4240697234","https://openalex.org/W1023722853","https://openalex.org/W4210563834","https://openalex.org/W4394006074","https://openalex.org/W2549579649","https://openalex.org/W2067804195"],"abstract_inverted_index":{"A":[0],"graph":[1,46,82,113,129],"$G$":[2],"is":[3,47,83,93,139],"det-extremal":[4,84,111,126],"if":[5,85,88],"$|\\operatorname{det}":[6],"A|=\\operatorname{per}":[7],"A$":[8],"for":[9],"its":[10,89],"adjacency":[11],"matrix":[12],"$A$.":[13],"Det-extremal":[14],"cubic":[15,60,81,105,112,127,150],"bipartite":[16],"graphs":[17,61,151],"arise":[18],"in":[19,159],"the":[20,29,37,48,59,160],"study":[21],"of":[22,34,36,44,51,58,162],"P\u00f3lya's":[23],"permanent":[24],"problem,":[25],"and":[26,86,107,135],"McCuaig":[27],"characterized":[28],"$3$-connected":[30,104],"ones":[31],"as":[32],"vertex-sums":[33],"copies":[35],"Heawood":[38],"graph.":[39],"The":[40],"total":[41,54,63,90,153],"domatic":[42,64,91,154],"number":[43,50,65,92,155],"a":[45,69,79,124,172],"largest":[49],"pairwise":[52],"disjoint":[53],"dominating":[55],"sets.":[56],"Characterization":[57],"with":[62,152],"$1$":[66],"has":[67,114,130,171],"been":[68],"long-standing":[70],"open":[71],"problem.":[72],"In":[73,157],"this":[74,137,143],"paper,":[75],"we":[76],"prove":[77,122],"that":[78,98,108,123,136,167],"connected":[80,110,125],"only":[87],"$1$.":[94,156],"We":[95,120],"further":[96],"show":[97],"McCuaig's":[99],"characterization":[100],"extends":[101],"to":[102,149],"all":[103],"graphs,":[106],"every":[109,168],"girth":[115],"$3$,":[116],"$5$":[117],"or":[118],"$6$.":[119],"also":[121],"non-bipartite":[128],"at":[131],"least":[132],"$28$":[133],"vertices,":[134],"bound":[138],"best":[140],"possible.":[141],"Through":[142],"correspondence,":[144],"these":[145],"results":[146,165],"carry":[147],"over":[148],"addition,":[158],"language":[161],"configurations,":[163],"our":[164],"imply":[166],"triangle-free":[169],"$3$-configuration":[170],"blocking":[173],"set.":[174]},"counts_by_year":[{"year":2026,"cited_by_count":2}],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-30T00:00:00"}
