{"id":"https://openalex.org/W7141659455","doi":"https://doi.org/10.48550/arxiv.2603.24880","title":"The Four Color Theorem with Linearly Many Reducible Configurations and Near-Linear Time Coloring","display_name":"The Four Color Theorem with Linearly Many Reducible Configurations and Near-Linear Time Coloring","publication_year":2026,"publication_date":"2026-03-25","ids":{"openalex":"https://openalex.org/W7141659455","doi":"https://doi.org/10.48550/arxiv.2603.24880"},"language":null,"primary_location":{"id":"doi:10.48550/arxiv.2603.24880","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2603.24880","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"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":"article"},"type":"preprint","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.48550/arxiv.2603.24880","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5102678203","display_name":"Yuta Inoue","orcid":null},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Inoue, Yuta","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5130801065","display_name":"Ken-ichi Kawarabayashi","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Kawarabayashi, Ken-ichi","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5104315325","display_name":"Atsuyuki Miyashita","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Miyashita, Atsuyuki","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5005993212","display_name":"Bojan Mohar","orcid":"https://orcid.org/0000-0002-7408-6148"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Mohar, Bojan","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5062275812","display_name":"Carsten Thomassen","orcid":"https://orcid.org/0000-0003-0670-4079"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Thomassen, Carsten","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5130753177","display_name":"Mikkel Thorup","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Thorup, Mikkel","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":6,"corresponding_author_ids":["https://openalex.org/A5102678203"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"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/T10996","display_name":"Computational Geometry and Mesh Generation","score":0.8762999773025513,"subfield":{"id":"https://openalex.org/subfields/1704","display_name":"Computer Graphics and Computer-Aided Design"},"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/T10996","display_name":"Computational Geometry and Mesh Generation","score":0.8762999773025513,"subfield":{"id":"https://openalex.org/subfields/1704","display_name":"Computer Graphics and Computer-Aided Design"},"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/T10374","display_name":"Advanced Graph Theory Research","score":0.03920000046491623,"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":0.023800000548362732,"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/mathematical-proof","display_name":"Mathematical proof","score":0.7649999856948853},{"id":"https://openalex.org/keywords/generalization","display_name":"Generalization","score":0.6144999861717224},{"id":"https://openalex.org/keywords/pairwise-comparison","display_name":"Pairwise comparison","score":0.569599986076355},{"id":"https://openalex.org/keywords/time-complexity","display_name":"Time complexity","score":0.5033000111579895},{"id":"https://openalex.org/keywords/quadratic-equation","display_name":"Quadratic equation","score":0.46059998869895935},{"id":"https://openalex.org/keywords/triangulation","display_name":"Triangulation","score":0.45489999651908875},{"id":"https://openalex.org/keywords/planar","display_name":"Planar","score":0.43639999628067017},{"id":"https://openalex.org/keywords/constant","display_name":"Constant (computer programming)","score":0.39469999074935913}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7782999873161316},{"id":"https://openalex.org/C108710211","wikidata":"https://www.wikidata.org/wiki/Q11538","display_name":"Mathematical proof","level":2,"score":0.7649999856948853},{"id":"https://openalex.org/C177148314","wikidata":"https://www.wikidata.org/wiki/Q170084","display_name":"Generalization","level":2,"score":0.6144999861717224},{"id":"https://openalex.org/C184898388","wikidata":"https://www.wikidata.org/wiki/Q1435712","display_name":"Pairwise comparison","level":2,"score":0.569599986076355},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5371999740600586},{"id":"https://openalex.org/C311688","wikidata":"https://www.wikidata.org/wiki/Q2393193","display_name":"Time complexity","level":2,"score":0.5033000111579895},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.46149998903274536},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.46059998869895935},{"id":"https://openalex.org/C135981907","wikidata":"https://www.wikidata.org/wiki/Q188056","display_name":"Triangulation","level":2,"score":0.45489999651908875},{"id":"https://openalex.org/C134786449","wikidata":"https://www.wikidata.org/wiki/Q3391255","display_name":"Planar","level":2,"score":0.43639999628067017},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.39469999074935913},{"id":"https://openalex.org/C184670325","wikidata":"https://www.wikidata.org/wiki/Q512604","display_name":"Loop (graph theory)","level":2,"score":0.3774999976158142},{"id":"https://openalex.org/C2778307483","wikidata":"https://www.wikidata.org/wiki/Q5149038","display_name":"Colored","level":2,"score":0.362199991941452},{"id":"https://openalex.org/C41045048","wikidata":"https://www.wikidata.org/wiki/Q202843","display_name":"Linear programming","level":2,"score":0.35109999775886536},{"id":"https://openalex.org/C195065555","wikidata":"https://www.wikidata.org/wiki/Q214881","display_name":"Curvature","level":2,"score":0.35040000081062317},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.33340001106262207},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.2888000011444092},{"id":"https://openalex.org/C4107886","wikidata":"https://www.wikidata.org/wiki/Q906377","display_name":"Discrete geometry","level":2,"score":0.2865999937057495},{"id":"https://openalex.org/C148043351","wikidata":"https://www.wikidata.org/wiki/Q4456944","display_name":"Current (fluid)","level":2,"score":0.2759999930858612},{"id":"https://openalex.org/C179799912","wikidata":"https://www.wikidata.org/wiki/Q205084","display_name":"Computational complexity theory","level":2,"score":0.263700008392334},{"id":"https://openalex.org/C101837359","wikidata":"https://www.wikidata.org/wiki/Q547823","display_name":"Planar graph","level":3,"score":0.25999999046325684}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.48550/arxiv.2603.24880","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2603.24880","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"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":null,"raw_source_name":null,"raw_type":"article"}],"best_oa_location":{"id":"doi:10.48550/arxiv.2603.24880","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2603.24880","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"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":"article"},"sustainable_development_goals":[{"display_name":"Sustainable cities and communities","id":"https://metadata.un.org/sdg/11","score":0.6136162877082825}],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"We":[0],"give":[1],"a":[2,89,160,189,201,218,239,250,260],"near-linear":[3],"time":[4,15],"4-coloring":[5,76,251],"algorithm":[6,16,25],"for":[7,73,168],"planar":[8,51,162],"graphs,":[9],"improving":[10],"on":[11,109,200],"the":[12,30,34,43,48,82,86,97,105,119,130,147,192,235],"previous":[13],"quadratic":[14,219],"by":[17,29,206,238],"Robertson":[18],"et":[19],"al.":[20],"from":[21],"1996.":[22],"Such":[23],"an":[24],"cannot":[26],"be":[27,270,274],"achieved":[28],"known":[31,79,131,286],"proofs":[32,80,132,194],"of":[33,47,65,81,88,133,184,226,263],"Four":[35],"Color":[36],"Theorem":[37],"(4CT).":[38],"Technically":[39],"speaking,":[40],"we":[41,137,153,174,232,266],"show":[42,85],"following":[44],"significant":[45],"generalization":[46],"4CT:":[49],"every":[50],"triangulation":[52,183],"contains":[53],"linearly":[54],"many":[55],"pairwise":[56,61],"non-touching":[57],"reducible":[58,91,114,227,264,281],"configurations":[59,115,228,282],"or":[60,93,229],"non-crossing":[62],"obstructing":[63,94,230],"cycles":[64],"length":[66],"at":[67],"most":[68],"5":[69],"(which":[70],"all":[71],"allow":[72],"making":[74],"effective":[75],"reductions).":[77],"The":[78,100,210],"4CT":[83],"only":[84],"existence":[87,101],"single":[90],"configuration":[92],"cycle":[95],"in":[96,116,142,159,180,217,252,276],"above":[98],"statement.":[99],"is":[102,149,204,285],"proved":[103],"using":[104],"discharging":[106],"method":[107],"based":[108],"combinatorial":[110,124],"curvature.":[111,125],"It":[112],"identifies":[113],"parts":[117,145,179],"where":[118,146],"local":[120],"neighborhood":[121],"has":[122],"positive":[123],"Our":[126,242],"result":[127],"significantly":[128],"strengthens":[129],"4CT,":[134],"showing":[135],"that":[136,203],"can":[138,154,175,233],"also":[139,165],"find":[140,176],"reductions":[141,156],"large":[143],"``flat\"":[144],"curvature":[148],"zero,":[150],"and":[151],"moreover,":[152],"make":[155],"almost":[157],"anywhere":[158],"given":[161],"graph.":[163],"This":[164],"opens":[166],"possibilities":[167],"extensions":[169],"to":[170,197,269,273],"higher":[171],"surfaces":[172],"since":[173],"such":[177],"flat":[178],"any":[181,185],"large-width":[182],"fixed":[186],"surface.":[187],"From":[188],"computational":[190],"perspective,":[191],"old":[193],"allowed":[195],"us":[196],"apply":[198],"induction":[199],"problem":[202,236],"smaller":[205],"some":[207],"additive":[208],"constant.":[209],"inductive":[211,243],"step":[212,244],"took":[213],"linear":[214,224,261],"time,":[215,248],"resulting":[216],"total":[220,255],"time.":[221,256],"With":[222],"our":[223,280],"number":[225,262],"cycles,":[231],"reduce":[234],"size":[237],"constant":[240],"factor.":[241],"takes":[245],"$O(n\\log":[246,253],"n)$":[247,254],"yielding":[249],"To":[257],"efficiently":[258],"handle":[259],"configurations,":[265],"need":[267],"them":[268],"sufficiently":[271],"robust":[272],"useful":[275],"other":[277],"applications.":[278],"All":[279],"are":[283],"what":[284],"as":[287],"D-reducible.":[288]},"counts_by_year":[],"updated_date":"2026-05-09T06:09:20.037420","created_date":"2026-03-28T00:00:00"}