{"id":"https://openalex.org/W7225799765","doi":"https://doi.org/10.5281/zenodo.24128770","title":"The Heawood family is not the complete list of excluded minors for Colin de Verdi\u00e8re invariant at most 5","display_name":"The Heawood family is not the complete list of excluded minors for Colin de Verdi\u00e8re invariant at most 5","publication_year":2026,"publication_date":"2026-11-11","ids":{"openalex":"https://openalex.org/W7225799765","doi":"https://doi.org/10.5281/zenodo.24128770"},"language":"en","primary_location":{"id":"doi:10.5281/zenodo.24128770","doi":"https://doi.org/10.5281/zenodo.24128770","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24128770","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.5281/zenodo.24128770","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5159121649","display_name":"P. Graham","orcid":"https://orcid.org/0009-0007-8493-1505","observed_orcids":["https://orcid.org/0009-0007-8493-1505"]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Graham, P.","raw_affiliation_strings":[],"raw_orcid":"https://orcid.org/0009-0007-8493-1505","affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5159121649"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"cited_by_count":1,"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":0.986299991607666,"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":0.986299991607666,"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":0.0031999999191612005,"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/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.0026000000070780516,"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":[],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.8399999737739563},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.696399986743927},{"id":"https://openalex.org/C2779760435","wikidata":"https://www.wikidata.org/wiki/Q5396169","display_name":"Minor (academic)","level":2,"score":0.633400022983551},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.5609999895095825},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5009999871253967},{"id":"https://openalex.org/C190470478","wikidata":"https://www.wikidata.org/wiki/Q2370229","display_name":"Invariant (physics)","level":2,"score":0.44359999895095825},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.421999990940094},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.35519999265670776},{"id":"https://openalex.org/C150591576","wikidata":"https://www.wikidata.org/wiki/Q739925","display_name":"Family of sets","level":3,"score":0.33009999990463257},{"id":"https://openalex.org/C125080357","wikidata":"https://www.wikidata.org/wiki/Q45715","display_name":"Complete graph","level":3,"score":0.29980000853538513},{"id":"https://openalex.org/C136979486","wikidata":"https://www.wikidata.org/wiki/Q773483","display_name":"Existential quantification","level":2,"score":0.2985000014305115}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.5281/zenodo.24128770","doi":"https://doi.org/10.5281/zenodo.24128770","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24128770","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},{"id":"doi:10.5281/zenodo.24128771","doi":"https://doi.org/10.5281/zenodo.24128771","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24128771","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"doi:10.5281/zenodo.24128770","doi":"https://doi.org/10.5281/zenodo.24128770","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24128770","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":5,"referenced_works":["https://openalex.org/W2121189168","https://openalex.org/W4242108867","https://openalex.org/W4402480977","https://openalex.org/W4402698778","https://openalex.org/W7167224464"],"related_works":[],"abstract_inverted_index":{"We":[0],"show":[1],"that":[2,76,116],"two":[3,119],"intrinsically":[4],"knotted":[5],"graphs":[6,33],"$G_1$":[7,77,90,109],"and":[8,15,24,37,42,64,72,78,91,99,110,134,164,168,189,199],"$G_2$":[9,79,111],"with":[10,96],"23":[11],"edges,":[12,101],"on":[13],"11":[14,97],"12":[16],"vertices,":[17],"have":[18,80],"Colin":[19],"de":[20],"Verdi\u00e8re":[21],"invariant":[22],"$\\mu=6$":[23],"no":[25],"minor":[26,87,105,170],"in":[27,162],"the":[28,31,45,50,85,117,139,143,169,186,190,203,217],"Heawood":[29,46,140],"family,":[30],"78":[32],"obtained":[34],"from":[35,74],"$K_7$":[36],"$K_{3,3,1,1}$":[38],"by":[39,158,172,182,194,210,213],"$\\Delta":[40],"Y$-":[41],"$Y\\Delta$-transformations.":[43],"So":[44],"family":[47,141],"is":[48,102,131,142,156],"not":[49,178,207],"complete":[51,144],"list":[52,145],"of":[53,89,121,146],"excluded":[54,104,147],"minors":[55,148],"for":[56,106,149],"$\\mu\\le5$,":[57,107],"as":[58,69],"van":[59,122],"der":[60,123],"Holst":[61,124],"proposed;":[62],"Mattman":[63],"Pavelescu":[65],"stated":[66],"this":[67],"proposal":[68],"a":[70,93,129],"conjecture":[71],"derived":[73],"it":[75],"$\\mu=5$.":[81],"More":[82],"precisely,":[83],"either":[84],"common":[86],"$H$":[88],"$G_2$,":[92],"4-regular":[94],"graph":[95,130],"vertices":[98],"22":[100],"an":[103],"or":[108],"both":[112,126],"are.":[113],"It":[114],"follows":[115],"following":[118],"conjectures":[120],"cannot":[125],"hold:":[127],"(a)":[128],"4-flat":[132,150],"if":[133,136],"only":[135],"$\\mu\\le5$;":[137],"(b)":[138],"graphs.":[151],"The":[152,184],"lower":[153],"bound":[154],"$\\mu\\ge6$":[155],"proved":[157],"explicit":[159],"matrices":[160],"certified":[161],"interval":[163],"exact":[165],"rational":[166],"arithmetic,":[167],"exclusion":[171],"exhaustive":[173],"enumeration.":[174],"This":[175],"work":[176],"has":[177],"yet":[179,208],"been":[180],"reviewed":[181,209],"experts.":[183,211],"proofs,":[185],"computer":[187],"checks":[188],"text":[191],"were":[192],"produced":[193],"AI":[195,214],"models":[196,215],"(from":[197],"OpenAI":[198],"Anthropic),":[200],"working":[201],"under":[202,216],"author's":[204,218],"direction.":[205,219],"Computer-assisted;":[206],"Produced":[212]},"counts_by_year":[{"year":2026,"cited_by_count":1}],"updated_date":"2026-11-12T12:28:57.059296","created_date":"2026-11-12T00:00:00"}