{"id":"https://openalex.org/W7225148368","doi":"https://doi.org/10.5281/zenodo.23915320","title":"A subdivided 6-clique breaks a conjectured sum bound for independent monophonic-position colorings","display_name":"A subdivided 6-clique breaks a conjectured sum bound for independent monophonic-position colorings","publication_year":2026,"publication_date":"2026-11-03","ids":{"openalex":"https://openalex.org/W7225148368","doi":"https://doi.org/10.5281/zenodo.23915320"},"language":"en","primary_location":{"id":"doi:10.5281/zenodo.23915320","doi":"https://doi.org/10.5281/zenodo.23915320","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23915320","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","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.5281/zenodo.23915320","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":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":0.9887999892234802,"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":0.9887999892234802,"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":0.007199999876320362,"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":0.0013000000035390258,"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/graph-complement","display_name":"graph complement","score":0.5059999823570251},{"id":"https://openalex.org/keywords/edge-subdivision","display_name":"edge subdivision","score":0.6779999732971191},{"id":"https://openalex.org/keywords/counterexample","display_name":"counterexample","score":0.6290000081062317},{"id":"https://openalex.org/keywords/formal-verification","display_name":"formal verification","score":0.7459999918937683},{"id":"https://openalex.org/keywords/lower-bounds","display_name":"lower bounds","score":0.34299999475479126}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7906000018119812},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6653000116348267},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.6198999881744385},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.5454999804496765},{"id":"https://openalex.org/C112313634","wikidata":"https://www.wikidata.org/wiki/Q7886648","display_name":"Complement (music)","level":5,"score":0.5383999943733215},{"id":"https://openalex.org/C108710211","wikidata":"https://www.wikidata.org/wiki/Q11538","display_name":"Mathematical proof","level":2,"score":0.5133000016212463},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.4950000047683716},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.45489999651908875},{"id":"https://openalex.org/C76946457","wikidata":"https://www.wikidata.org/wiki/Q504843","display_name":"Graph coloring","level":3,"score":0.4002000093460083},{"id":"https://openalex.org/C123809776","wikidata":"https://www.wikidata.org/wiki/Q1050972","display_name":"Edge coloring","level":5,"score":0.3937999904155731},{"id":"https://openalex.org/C122818955","wikidata":"https://www.wikidata.org/wiki/Q1060343","display_name":"Independent set","level":3,"score":0.38600000739097595},{"id":"https://openalex.org/C21642379","wikidata":"https://www.wikidata.org/wiki/Q3342988","display_name":"Complete coloring","level":5,"score":0.3614000082015991},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.34940001368522644},{"id":"https://openalex.org/C196956537","wikidata":"https://www.wikidata.org/wiki/Q202021","display_name":"Chromatic scale","level":2,"score":0.3481000065803528},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.3361999988555908},{"id":"https://openalex.org/C158319403","wikidata":"https://www.wikidata.org/wiki/Q5477527","display_name":"Fractional coloring","level":5,"score":0.32499998807907104},{"id":"https://openalex.org/C168291704","wikidata":"https://www.wikidata.org/wiki/Q902252","display_name":"Complement graph","level":5,"score":0.2896000146865845},{"id":"https://openalex.org/C311688","wikidata":"https://www.wikidata.org/wiki/Q2393193","display_name":"Time complexity","level":2,"score":0.26179999113082886},{"id":"https://openalex.org/C159254197","wikidata":"https://www.wikidata.org/wiki/Q1144915","display_name":"Lexicographical order","level":2,"score":0.2574999928474426},{"id":"https://openalex.org/C2777735758","wikidata":"https://www.wikidata.org/wiki/Q817765","display_name":"Path (computing)","level":2,"score":0.251800000667572}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.5281/zenodo.23915320","doi":"https://doi.org/10.5281/zenodo.23915320","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23915320","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.23915321","doi":"https://doi.org/10.5281/zenodo.23915321","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23915321","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.23915320","doi":"https://doi.org/10.5281/zenodo.23915320","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23915320","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":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":17,"referenced_works":["https://openalex.org/W2100609826","https://openalex.org/W2319043422","https://openalex.org/W2972824562","https://openalex.org/W3113382290","https://openalex.org/W3134775563","https://openalex.org/W3137055701","https://openalex.org/W3170509085","https://openalex.org/W4392236482","https://openalex.org/W4392958912","https://openalex.org/W4394789299","https://openalex.org/W4403375082","https://openalex.org/W4405433079","https://openalex.org/W4416192020","https://openalex.org/W7002039157","https://openalex.org/W7008161220","https://openalex.org/W7125798805","https://openalex.org/W7203703910"],"related_works":["https://openalex.org/W3113382290","https://openalex.org/W7099630476","https://openalex.org/W7133452956","https://openalex.org/W2037509864","https://openalex.org/W2622711670","https://openalex.org/W2608830549","https://openalex.org/W7095164665","https://openalex.org/W7133384651","https://openalex.org/W7206163023","https://openalex.org/W3217611830"],"abstract_inverted_index":{"A":[0],"set":[1],"of":[2,4,17,58,63,112,155],"vertices":[3,124],"a":[5,83,119,179],"graph":[6,61,115,120],"is":[7,144],"in":[8],"monophonic":[9,105],"position":[10],"if":[11],"no":[12],"induced":[13],"path":[14],"contains":[15],"three":[16],"its":[18,64,135],"vertices.":[19,99],"Ullas":[20],"Chandran":[21],"S.":[22,27],"V.,":[23],"Gabriele":[24],"Di":[25],"Stefano,":[26],"Haritha,":[28],"Elias":[29],"John":[30],"Thomas":[31],"and":[32,43,62,71,157,196,206],"James":[33],"Tuite":[34],"introduced":[35],"colorings":[36],"whose":[37,46],"color":[38,132,163],"classes":[39,47,164],"are":[40,48,171],"such":[41],"sets,":[42],"the":[44,54,73,78,103,113,142,151,194,197,210,224],"variant":[45],"also":[49,172],"independent.":[50],"They":[51],"showed":[52],"that":[53,102,125,161],"monophonic-position":[55,131,153],"chromatic":[56],"numbers":[57],"an":[59],"$n$-vertex":[60],"complement":[65,136],"sum":[66,106,143],"to":[67],"at":[68,96,127,138,145],"most":[69,97],"$n+1$,":[70],"conjectured":[72],"same":[74],"Nordhaus\u2013Gaddum":[75],"bound":[76,107],"for":[77,92],"independent":[79,104,130,152],"variant;":[80],"they":[81],"report":[82],"computer":[84],"search":[85],"by":[86,159,174,190,201,217,220],"Grahame":[87],"Erskine":[88],"confirming":[89],"their":[90],"conjecture":[91],"all":[93],"graphs":[94],"with":[95,178],"ten":[98],"We":[100],"show":[101],"fails.":[108],"Subdividing":[109],"every":[110],"edge":[111],"complete":[114],"$K_6$":[116],"once":[117],"gives":[118],"$H$":[121,156],"on":[122],"21":[123],"needs":[126,137],"least":[128,139,146],"eight":[129],"classes,":[133],"while":[134],"fifteen,":[140],"so":[141],"23.":[147],"The":[148,168,192],"proof":[149],"classifies":[150],"sets":[154],"shows":[158],"counting":[160],"seven":[162],"cannot":[165],"cover":[166],"$H$.":[167],"lower":[169],"bounds":[170],"confirmed":[173],"SAT-solver":[175],"proofs":[176],"checked":[177],"formally":[180],"verified":[181],"checker.":[182],"This":[183],"work":[184],"has":[185],"not":[186,214],"yet":[187,215],"been":[188],"reviewed":[189,216],"experts.":[191,218],"proofs,":[193],"code":[195],"text":[198],"were":[199],"produced":[200],"AI":[202,221],"models":[203,222],"(from":[204],"OpenAI":[205],"Anthropic),":[207],"working":[208],"under":[209,223],"author's":[211,225],"direction.":[212,226],"Computer-assisted;":[213],"Produced":[219]},"counts_by_year":[],"updated_date":"2026-11-05T12:25:35.769452","created_date":"2026-11-04T00:00:00"}
