{"id":"https://openalex.org/W7214758478","doi":"https://doi.org/10.48550/arxiv.2609.33035","title":"There are no nontrivial uniquely C_9-saturated graphs","display_name":"There are no nontrivial uniquely C_9-saturated graphs","publication_year":2026,"publication_date":"2026-09-27","ids":{"openalex":"https://openalex.org/W7214758478","doi":"https://doi.org/10.48550/arxiv.2609.33035"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.33035","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.33035","pdf_url":"https://arxiv.org/pdf/2609.33035","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/2609.33035","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5151085917","display_name":"Fayzan Khan","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Khan, Fayzan","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5151085917"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"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":0.5985999703407288,"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":0.5985999703407288,"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/T11329","display_name":"Limits and Structures in Graph Theory","score":0.39089998602867126,"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/T11476","display_name":"Graph theory and applications","score":0.0017000000225380063,"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/even-cycles","display_name":"even cycles","score":0.597000002861023},{"id":"https://openalex.org/keywords/graph-complement","display_name":"graph complement","score":0.43299999833106995},{"id":"https://openalex.org/keywords/graph-enumeration","display_name":"graph enumeration","score":0.41200000047683716},{"id":"https://openalex.org/keywords/computer-assisted-proof","display_name":"computer-assisted proof","score":0.5730000138282776}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8080000281333923},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.770799994468689},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7127000093460083},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.5727999806404114},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5519000291824341},{"id":"https://openalex.org/C2777759810","wikidata":"https://www.wikidata.org/wiki/Q149316","display_name":"Lemma (botany)","level":3,"score":0.5230000019073486},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.4560999870300293},{"id":"https://openalex.org/C184264201","wikidata":"https://www.wikidata.org/wiki/Q21199","display_name":"Natural number","level":2,"score":0.366100013256073},{"id":"https://openalex.org/C2780813799","wikidata":"https://www.wikidata.org/wiki/Q3274237","display_name":"Zero (linguistics)","level":2,"score":0.34279999136924744},{"id":"https://openalex.org/C156340839","wikidata":"https://www.wikidata.org/wiki/Q2704791","display_name":"Enumeration","level":2,"score":0.33820000290870667},{"id":"https://openalex.org/C88230418","wikidata":"https://www.wikidata.org/wiki/Q131476","display_name":"Graph theory","level":2,"score":0.3147999942302704},{"id":"https://openalex.org/C162307627","wikidata":"https://www.wikidata.org/wiki/Q204833","display_name":"Enhanced Data Rates for GSM Evolution","level":2,"score":0.2766000032424927},{"id":"https://openalex.org/C17231256","wikidata":"https://www.wikidata.org/wiki/Q5156540","display_name":"Completeness (order theory)","level":2,"score":0.25540000200271606},{"id":"https://openalex.org/C162392398","wikidata":"https://www.wikidata.org/wiki/Q272404","display_name":"Finite set","level":2,"score":0.2540000081062317},{"id":"https://openalex.org/C146380142","wikidata":"https://www.wikidata.org/wiki/Q1137726","display_name":"Directed graph","level":2,"score":0.2531999945640564}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.33035","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.33035","pdf_url":"https://arxiv.org/pdf/2609.33035","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.2609.33035","doi":"https://doi.org/10.48550/arxiv.2609.33035","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.33035","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":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.33035","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.33035","pdf_url":"https://arxiv.org/pdf/2609.33035","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":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7214758478.pdf","grobid_xml":"https://content.openalex.org/works/W7214758478.grobid-xml"},"referenced_works_count":1,"referenced_works":["https://openalex.org/W2258821783"],"related_works":["https://openalex.org/W2952286798","https://openalex.org/W1490824980","https://openalex.org/W2023305347","https://openalex.org/W3009029689","https://openalex.org/W7099037202","https://openalex.org/W2006394242","https://openalex.org/W1974941919","https://openalex.org/W3190516727","https://openalex.org/W7203883644","https://openalex.org/W4388572658"],"abstract_inverted_index":{"A":[0],"graph":[1],"G":[2,7],"is":[3,31,84,92,111],"uniquely":[4,47,87],"C_t-saturated":[5],"if":[6,33],"contains":[8,23],"no":[9,45,85],"cycle":[10,26,104,139],"of":[11,19,27,100,105,134,148,178],"length":[12,28,98,106],"t":[13,38,59,71],"and,":[14],"for":[15,57,81,128],"every":[16,58],"edge":[17],"e":[18],"the":[20,55,62,79,97,135,138,168,175],"complement,":[21],"G+e":[22],"exactly":[24],"one":[25],"t;":[29],"it":[30,34],"nontrivial":[32,46,86],"has":[35,74],"at":[36,107],"least":[37],"vertices.":[39],"Wenger":[40,182],"and":[41,53,70,122,130,160,183],"West":[42,184],"proved":[43],"that":[44],"C_6-":[48],"or":[49,116],"C_7-saturated":[50],"graphs":[51],"exist,":[52],"conjectured":[54],"same":[56],">=":[60,72],"6;":[61],"case":[63,94,142],"t=8":[64],"was":[65],"verified":[66],"but":[67],"never":[68],"published,":[69],"9":[73],"remained":[75],"open.":[76],"We":[77],"prove":[78],"conjecture":[80],"t=9:":[82],"there":[83],"C_9-saturated":[88],"graph.":[89],"The":[90,118],"proof":[91],"a":[93,101,144,152,179],"analysis":[95],"on":[96],"L":[99],"longest":[102],"even":[103],"most":[108],"12":[109],"(L":[110],"4,":[112],"6,":[113],"8,":[114],"10,":[115],"12).":[117],"cases":[119],"L=12,":[120],"L=10,":[121],"L=4":[123],"are":[124],"settled":[125],"by":[126,151,163],"hand;":[127],"L=6":[129],"L=8,":[131],"hand":[132],"classifications":[133],"components":[136],"outside":[137],"reduce":[140],"each":[141],"to":[143],"bounded":[145],"finite":[146],"family":[147],"configurations,":[149],"eliminated":[150],"short,":[153],"replayable":[154],"computer":[155],"enumeration":[156],"with":[157],"known-answer":[158],"controls":[159],"completeness":[161],"certified":[162],"zero":[164],"cap":[165],"hits.":[166],"Along":[167],"way":[169],"we":[170],"prove,":[171],"in":[172],"sharpened":[173],"form,":[174],"t=9":[176],"instance":[177],"structural":[180],"lemma":[181],"stated":[185],"without":[186],"proof.":[187]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-30T00:00:00"}