{"id":"https://openalex.org/W7208727863","doi":"https://doi.org/10.48550/arxiv.2609.03566","title":"Cohen--Macaulayness of $\\mathrm{Inc}(\\mathbb{N})$-Invariant Chains of Edge Ideals","display_name":"Cohen--Macaulayness of $\\mathrm{Inc}(\\mathbb{N})$-Invariant Chains of Edge Ideals","publication_year":2026,"publication_date":"2026-09-03","ids":{"openalex":"https://openalex.org/W7208727863","doi":"https://doi.org/10.48550/arxiv.2609.03566"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.03566","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.03566","pdf_url":"https://arxiv.org/pdf/2609.03566","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":"public-domain","license_id":"https://openalex.org/licenses/public-domain","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2609.03566","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5039084348","display_name":"Imran Anwar","orcid":"https://orcid.org/0000-0002-9043-4333","observed_orcids":["https://orcid.org/0000-0002-9043-4333"]},"institutions":[{"id":"https://openalex.org/I207789805","display_name":"Lahore University of Management Sciences","ror":"https://ror.org/05b5x4a35","country_code":"PK","type":"education","lineage":["https://openalex.org/I207789805"]}],"countries":["PK"],"is_corresponding":false,"raw_author_name":"Anwar, Imran","raw_affiliation_strings":["Department of Mathematics , Lahore University of Management Sciences , Lahore ,"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics , Lahore University of Management Sciences , Lahore ,","institution_ids":["https://openalex.org/I207789805"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5117437462","display_name":"Mughees Ghayas","orcid":"https://orcid.org/0009-0005-2177-2354","observed_orcids":["https://orcid.org/0009-0005-2177-2354"]},"institutions":[{"id":"https://openalex.org/I207789805","display_name":"Lahore University of Management Sciences","ror":"https://ror.org/05b5x4a35","country_code":"PK","type":"education","lineage":["https://openalex.org/I207789805"]}],"countries":["PK"],"is_corresponding":false,"raw_author_name":"Ghayas, Mughees","raw_affiliation_strings":["Department of Mathematics , Lahore University of Management Sciences , Lahore ,","Pak- istan Department of Mathematics , Lahore University of Management Sciences , Lahore , Pak- istan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics , Lahore University of Management Sciences , Lahore ,","institution_ids":["https://openalex.org/I207789805"]},{"raw_affiliation_string":"Pak- istan Department of Mathematics , Lahore University of Management Sciences , Lahore , Pak- istan","institution_ids":["https://openalex.org/I207789805"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5149987467","display_name":"Azhar Javed","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I207789805","display_name":"Lahore University of Management Sciences","ror":"https://ror.org/05b5x4a35","country_code":"PK","type":"education","lineage":["https://openalex.org/I207789805"]}],"countries":["PK"],"is_corresponding":false,"raw_author_name":"Javed, Azhar","raw_affiliation_strings":["Department of Mathematics , Lahore University of Management Sciences , Lahore ,"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics , Lahore University of Management Sciences , Lahore ,","institution_ids":["https://openalex.org/I207789805"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I207789805"],"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/T12069","display_name":"Commutative Algebra and Its Applications","score":0.9994999766349792,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T12069","display_name":"Commutative Algebra and Its Applications","score":0.9994999766349792,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"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/T11703","display_name":"Rings, Modules, and Algebras","score":0.0,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"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/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.0,"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/cohen-macaulay-rings","display_name":"Cohen-Macaulay rings","score":0.9559999704360962},{"id":"https://openalex.org/keywords/edge-ideals","display_name":"edge ideals","score":0.9940000176429749},{"id":"https://openalex.org/keywords/line-graphs","display_name":"line graphs","score":0.6779999732971191},{"id":"https://openalex.org/keywords/complete-graphs","display_name":"complete graphs","score":0.9760000109672546},{"id":"https://openalex.org/keywords/cyclic-graphs","display_name":"cyclic graphs","score":0.9909999966621399},{"id":"https://openalex.org/keywords/krull-dimension","display_name":"Krull dimension","score":0.8309999704360962},{"id":"https://openalex.org/keywords/height-of-ideals","display_name":"height of ideals","score":0.7059999704360962},{"id":"https://openalex.org/keywords/monomial-ideals","display_name":"monomial ideals","score":0.7129999995231628}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7771000266075134},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7142999768257141},{"id":"https://openalex.org/C160446614","wikidata":"https://www.wikidata.org/wiki/Q1322892","display_name":"Chordal graph","level":3,"score":0.5666999816894531},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.49900001287460327},{"id":"https://openalex.org/C74133993","wikidata":"https://www.wikidata.org/wiki/Q3115472","display_name":"Indifference graph","level":3,"score":0.4959999918937683},{"id":"https://openalex.org/C102192266","wikidata":"https://www.wikidata.org/wiki/Q4545823","display_name":"1-planar graph","level":4,"score":0.4925999939441681},{"id":"https://openalex.org/C10325715","wikidata":"https://www.wikidata.org/wiki/Q7835582","display_name":"Trapezoid graph","level":5,"score":0.47589999437332153},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.461899995803833},{"id":"https://openalex.org/C43517604","wikidata":"https://www.wikidata.org/wiki/Q7144893","display_name":"Pathwidth","level":4,"score":0.46140000224113464},{"id":"https://openalex.org/C186942658","wikidata":"https://www.wikidata.org/wiki/Q5134410","display_name":"Clique-sum","level":5,"score":0.4528999924659729},{"id":"https://openalex.org/C162307627","wikidata":"https://www.wikidata.org/wiki/Q204833","display_name":"Enhanced Data Rates for GSM Evolution","level":2,"score":0.4104999899864197},{"id":"https://openalex.org/C198352243","wikidata":"https://www.wikidata.org/wiki/Q37105","display_name":"Line (geometry)","level":2,"score":0.3817000091075897},{"id":"https://openalex.org/C203776342","wikidata":"https://www.wikidata.org/wiki/Q1378376","display_name":"Line graph","level":3,"score":0.35569998621940613},{"id":"https://openalex.org/C199185054","wikidata":"https://www.wikidata.org/wiki/Q552299","display_name":"Chain (unit)","level":2,"score":0.3422999978065491},{"id":"https://openalex.org/C59824394","wikidata":"https://www.wikidata.org/wiki/Q5141281","display_name":"Cograph","level":5,"score":0.33239999413490295},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.3292999863624573},{"id":"https://openalex.org/C187407849","wikidata":"https://www.wikidata.org/wiki/Q6889712","display_name":"Modular decomposition","level":5,"score":0.32339999079704285},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.3188000023365021},{"id":"https://openalex.org/C18359143","wikidata":"https://www.wikidata.org/wiki/Q7888149","display_name":"Maximal independent set","level":5,"score":0.31540000438690186},{"id":"https://openalex.org/C68103157","wikidata":"https://www.wikidata.org/wiki/Q569347","display_name":"Graph product","level":5,"score":0.29019999504089355},{"id":"https://openalex.org/C8554925","wikidata":"https://www.wikidata.org/wiki/Q3893853","display_name":"Split graph","level":5,"score":0.27300000190734863},{"id":"https://openalex.org/C88230418","wikidata":"https://www.wikidata.org/wiki/Q131476","display_name":"Graph theory","level":2,"score":0.2574999928474426}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.03566","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.03566","pdf_url":"https://arxiv.org/pdf/2609.03566","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":"public-domain","license_id":"https://openalex.org/licenses/public-domain","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2609.03566","doi":"https://doi.org/10.48550/arxiv.2609.03566","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.03566","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":"public-domain","license_id":"https://openalex.org/licenses/public-domain","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.03566","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.03566","pdf_url":"https://arxiv.org/pdf/2609.03566","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":"public-domain","license_id":"https://openalex.org/licenses/public-domain","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320332162","display_name":"Centers for Disease Control and Prevention","ror":"https://ror.org/042twtr12"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7208727863.pdf","grobid_xml":"https://content.openalex.org/works/W7208727863.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W2099060864","https://openalex.org/W2295974917","https://openalex.org/W2291090077","https://openalex.org/W7115790826","https://openalex.org/W2004834330","https://openalex.org/W1750553821","https://openalex.org/W4232607766","https://openalex.org/W2799299041","https://openalex.org/W4200631314","https://openalex.org/W2058562264"],"abstract_inverted_index":{"Let":[0],"$(G_n)_{n\\ge":[1],"n_0}$":[2],"be":[3],"a":[4,73,155],"family":[5],"of":[6,40,52,63,70,88,95],"graphs":[7,37,42,45,48,54,58,133],"on":[8],"$[n]$":[9],"whose":[10],"edge":[11],"ideals":[12],"$I_n\\subseteq":[13],"R_n=k[x_1,\\dots,x_n]$":[14],"form":[15],"an":[16],"$\\mathrm{Inc}(\\mathbb{N})$-invariant":[17,121],"chain,":[18],"$I_{n+r}=\\mathrm{Inc}(\\mathbb{N})_{n,n+r}(I_n)$":[19],"for":[20,32,76,105],"$n\\ge":[21],"n_0,\\":[22],"r\\ge0$.":[23],"We":[24],"determine":[25],"which":[26,119],"pairs":[27],"$(n,r)$":[28],"make":[29],"$R_{n+r}/I_{n+r}$":[30,78],"Cohen--Macaulay,":[31],"five":[33],"classical":[34],"families:":[35],"line":[36,41,57,89],"$L_n$,":[38],"complements":[39,51,87,94],"$L_n^c$,":[43],"complete":[44,74,91,156],"$K_n$,":[46],"cyclic":[47,53,96,132],"$C_n$,":[49],"and":[50,67,72,82,93,110,145,151,161,165],"$C_n^c$.":[55],"For":[56,86,131],"we":[59,98,134],"give":[60],"the":[61,65,100,108,114,120,124],"generators":[62],"$I_{n+r}$,":[64],"height":[66],"Krull":[68],"dimension":[69],"$R_{n+r}/I_{n+r}$,":[71],"classification:":[75,157],"$n\\ge5$,":[77],"is":[79,102],"Cohen--Macaulay":[80,103],"if":[81,84,160,163],"only":[83,162],"$r=n-4$.":[85],"graphs,":[90,92,97],"show":[99],"chain":[101,122],"unconditionally,":[104],"every":[106],"$r\\ge0$;":[107],"first":[109],"third":[111],"arise":[112],"from":[113],"same":[115],"underlying":[116],"phenomenon,":[117],"in":[118,142],"reproduces":[123],"original":[125],"graph":[126],"itself":[127],"at":[128],"each":[129],"step.":[130],"prove":[135],"$\\mathrm{Inc}(\\mathbb{N})_{n,n+r}(I(C_n))=I(K_{n+r})$":[136],"once":[137],"$r\\ge":[138],"n-3$,":[139],"giving":[140],"Cohen--Macaulayness":[141,158],"this":[143],"range,":[144],"conjecture":[146],"--":[147,154],"with":[148],"supporting":[149],"computational":[150],"structural":[152],"evidence":[153],"holds":[159],"$r\\ge\\lfloor(n-4)/2\\rfloor$":[164],"$r\\ne":[166],"n-4$.":[167]},"counts_by_year":[],"updated_date":"2026-10-28T10:33:24.094632","created_date":"2026-09-05T00:00:00"}
