{"id":"https://openalex.org/W7214181582","doi":"https://doi.org/10.48550/arxiv.2609.27168","title":"Smallest Cubic Non-1-Planar Graphs","display_name":"Smallest Cubic Non-1-Planar Graphs","publication_year":2026,"publication_date":"2026-09-23","ids":{"openalex":"https://openalex.org/W7214181582","doi":"https://doi.org/10.48550/arxiv.2609.27168"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.27168","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.27168","pdf_url":"https://arxiv.org/pdf/2609.27168","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/2609.27168","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5058217001","display_name":"Sergey Pupyrev","orcid":"https://orcid.org/0000-0003-4089-673X","observed_orcids":["https://orcid.org/0000-0003-4089-673X"]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Pupyrev, Sergey","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5058217001"],"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":0.9932000041007996,"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":0.9932000041007996,"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/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.0015999999595806003,"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/T12541","display_name":"Graph Labeling and Dimension Problems","score":0.0005000000237487257,"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/1-planar-graphs","display_name":"1-planar graphs","score":0.9860000014305115},{"id":"https://openalex.org/keywords/cubic-graphs","display_name":"cubic graphs","score":0.9580000042915344},{"id":"https://openalex.org/keywords/girth","display_name":"girth","score":0.8119999766349792},{"id":"https://openalex.org/keywords/subcubic-graphs","display_name":"subcubic graphs","score":0.9430000185966492},{"id":"https://openalex.org/keywords/graph-automorphisms","display_name":"graph automorphisms","score":0.9369999766349792},{"id":"https://openalex.org/keywords/network-visualization","display_name":"network visualization","score":0.628000020980835},{"id":"https://openalex.org/keywords/computer-assisted-proof","display_name":"computer-assisted proof","score":0.6209999918937683}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7570000290870667},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7448999881744385},{"id":"https://openalex.org/C102192266","wikidata":"https://www.wikidata.org/wiki/Q4545823","display_name":"1-planar graph","level":4,"score":0.5328999757766724},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5066999793052673},{"id":"https://openalex.org/C74133993","wikidata":"https://www.wikidata.org/wiki/Q3115472","display_name":"Indifference graph","level":3,"score":0.4745999872684479},{"id":"https://openalex.org/C100500283","wikidata":"https://www.wikidata.org/wiki/Q1374495","display_name":"Cubic graph","level":5,"score":0.43299999833106995},{"id":"https://openalex.org/C59824394","wikidata":"https://www.wikidata.org/wiki/Q5141281","display_name":"Cograph","level":5,"score":0.42750000953674316},{"id":"https://openalex.org/C43517604","wikidata":"https://www.wikidata.org/wiki/Q7144893","display_name":"Pathwidth","level":4,"score":0.4253000020980835},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.4244000017642975},{"id":"https://openalex.org/C8554925","wikidata":"https://www.wikidata.org/wiki/Q3893853","display_name":"Split graph","level":5,"score":0.4162999987602234},{"id":"https://openalex.org/C160446614","wikidata":"https://www.wikidata.org/wiki/Q1322892","display_name":"Chordal graph","level":3,"score":0.40149998664855957},{"id":"https://openalex.org/C14201764","wikidata":"https://www.wikidata.org/wiki/Q1205074","display_name":"Symmetric graph","level":5,"score":0.39100000262260437},{"id":"https://openalex.org/C122818955","wikidata":"https://www.wikidata.org/wiki/Q1060343","display_name":"Independent set","level":3,"score":0.37630000710487366},{"id":"https://openalex.org/C203776342","wikidata":"https://www.wikidata.org/wiki/Q1378376","display_name":"Line graph","level":3,"score":0.3686999976634979},{"id":"https://openalex.org/C206815938","wikidata":"https://www.wikidata.org/wiki/Q7077445","display_name":"Odd graph","level":5,"score":0.3449999988079071},{"id":"https://openalex.org/C43413943","wikidata":"https://www.wikidata.org/wiki/Q4927193","display_name":"Block graph","level":5,"score":0.3391000032424927},{"id":"https://openalex.org/C7036158","wikidata":"https://www.wikidata.org/wiki/Q3385162","display_name":"Graph homomorphism","level":5,"score":0.33869999647140503},{"id":"https://openalex.org/C114727528","wikidata":"https://www.wikidata.org/wiki/Q7840044","display_name":"Triangle-free graph","level":5,"score":0.2996000051498413},{"id":"https://openalex.org/C123809776","wikidata":"https://www.wikidata.org/wiki/Q1050972","display_name":"Edge coloring","level":5,"score":0.29750001430511475},{"id":"https://openalex.org/C186942658","wikidata":"https://www.wikidata.org/wiki/Q5134410","display_name":"Clique-sum","level":5,"score":0.2935999929904938},{"id":"https://openalex.org/C68103157","wikidata":"https://www.wikidata.org/wiki/Q569347","display_name":"Graph product","level":5,"score":0.2874999940395355},{"id":"https://openalex.org/C76946457","wikidata":"https://www.wikidata.org/wiki/Q504843","display_name":"Graph coloring","level":3,"score":0.28380000591278076},{"id":"https://openalex.org/C18359143","wikidata":"https://www.wikidata.org/wiki/Q7888149","display_name":"Maximal independent set","level":5,"score":0.2833999991416931},{"id":"https://openalex.org/C125792925","wikidata":"https://www.wikidata.org/wiki/Q3115604","display_name":"Outerplanar graph","level":5,"score":0.2757999897003174},{"id":"https://openalex.org/C4255713","wikidata":"https://www.wikidata.org/wiki/Q7291988","display_name":"Random regular graph","level":5,"score":0.2703999876976013},{"id":"https://openalex.org/C101837359","wikidata":"https://www.wikidata.org/wiki/Q547823","display_name":"Planar graph","level":3,"score":0.262800008058548},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.26179999113082886},{"id":"https://openalex.org/C166911692","wikidata":"https://www.wikidata.org/wiki/Q7130424","display_name":"Pancyclic graph","level":5,"score":0.2556000053882599},{"id":"https://openalex.org/C45374587","wikidata":"https://www.wikidata.org/wiki/Q12525525","display_name":"Computation","level":2,"score":0.2513999938964844}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.27168","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.27168","pdf_url":"https://arxiv.org/pdf/2609.27168","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.2609.27168","doi":"https://doi.org/10.48550/arxiv.2609.27168","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.27168","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","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.27168","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.27168","pdf_url":"https://arxiv.org/pdf/2609.27168","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":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7214181582.pdf","grobid_xml":"https://content.openalex.org/works/W7214181582.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W2063563598","https://openalex.org/W99328947","https://openalex.org/W2126913730","https://openalex.org/W1502731484","https://openalex.org/W2133605239","https://openalex.org/W2922367700","https://openalex.org/W7169596418","https://openalex.org/W2986839545","https://openalex.org/W2888394699","https://openalex.org/W2058139201"],"abstract_inverted_index":{"A":[0],"graph":[1,35,41,53,100],"is":[2,13,59,63,71],"1-planar":[3,138,146],"if":[4],"it":[5],"has":[6],"a":[7,40,82,86,95],"drawing":[8],"in":[9,108,136],"which":[10],"every":[11,125],"edge":[12],"crossed":[14],"at":[15,128],"most":[16,129],"once.":[17],"We":[18,140],"show":[19,111],"that":[20,45,112],"the":[21,33,48,77,121],"smallest":[22],"cubic":[23,166],"non-1-planar":[24],"graphs":[25,31,70,116,150,167,173],"have":[26],"$30$":[27,57],"vertices.":[28],"Two":[29],"such":[30],"are":[32,117],"Tutte-Coxeter":[34],"of":[36,42,76,123,127,148,153,161,174],"girth":[37,43],"eight":[38],"and":[39,94],"seven":[44],"we":[46,80,119],"call":[47],"Byte":[49],"graph.":[50],"Every":[51],"subcubic":[52,115],"with":[54,85,168],"fewer":[55,172],"than":[56,163],"vertices":[58],"1-planar.":[60],"Our":[61],"proof":[62],"computer-assisted,":[64],"but":[65],"directly":[66],"testing":[67,160],"all":[68,113],"relevant":[69],"impractical.":[72],"To":[73,110],"establish":[74],"non-1-planarity":[75],"two":[78],"graphs,":[79],"extend":[81],"SAT-based":[83],"solver":[84],"custom":[87],"clause":[88],"propagator":[89],"based":[90,98],"on":[91,99,170],"separating":[92],"cycles":[93],"case":[96],"split":[97],"automorphisms,":[101],"allowing":[102],"independent":[103],"cases":[104],"to":[105,144],"be":[106],"solved":[107],"parallel.":[109],"smaller":[114,154,175],"1-planar,":[118],"introduce":[120],"concept":[122],"$k$-flexibility:":[124],"set":[126],"$k$":[130],"prescribed":[131],"edges":[132],"can":[133],"remain":[134],"uncrossed":[135],"some":[137],"drawing.":[139],"use":[141],"this":[142],"property":[143],"reconstruct":[145],"drawings":[147,152],"larger":[149],"from":[151],"$k$-flexible":[155],"graphs.":[156],"This":[157],"replaces":[158],"exhaustive":[159],"more":[162],"forty":[164],"billion":[165],"computations":[169],"far":[171],"order.":[176]},"counts_by_year":[],"updated_date":"2026-10-28T10:33:24.094632","created_date":"2026-09-25T00:00:00"}
