{"id":"https://openalex.org/W4411450266","doi":"https://doi.org/10.37236/13653","title":"A Variable Version of the Quasi-Kernel Conjecture","display_name":"A Variable Version of the Quasi-Kernel Conjecture","publication_year":2025,"publication_date":"2025-06-19","ids":{"openalex":"https://openalex.org/W4411450266","doi":"https://doi.org/10.37236/13653"},"language":"en","primary_location":{"id":"doi:10.37236/13653","is_oa":true,"landing_page_url":"https://doi.org/10.37236/13653","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v32i2p53/pdf","source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v32i2p53/pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5091768348","display_name":"Jiangdong Ai","orcid":"https://orcid.org/0000-0002-5691-3365"},"institutions":[{"id":"https://openalex.org/I205237279","display_name":"Nankai University","ror":"https://ror.org/01y1kjr75","country_code":"CN","type":"education","lineage":["https://openalex.org/I205237279"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Jiangdong Ai","raw_affiliation_strings":["Nankai University"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Nankai University","institution_ids":["https://openalex.org/I205237279"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5104324455","display_name":"Xiangzhou Liu","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Xiangzhou Liu","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5104357027","display_name":"Fei Peng","orcid":"https://orcid.org/0009-0001-1408-0553"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Fei Peng","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.14652106,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"32","issue":"2","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":0.9994999766349792,"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":0.9994999766349792,"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":0.9994000196456909,"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/T11151","display_name":"Advanced Topology and Set Theory","score":0.9923999905586243,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"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/conjecture","display_name":"Conjecture","score":0.8999511003494263},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.7967895269393921},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7694205045700073},{"id":"https://openalex.org/keywords/digraph","display_name":"Digraph","score":0.7385308146476746},{"id":"https://openalex.org/keywords/kernel","display_name":"Kernel (algebra)","score":0.6289010047912598},{"id":"https://openalex.org/keywords/vertex","display_name":"Vertex (graph theory)","score":0.43139469623565674},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4202047884464264},{"id":"https://openalex.org/keywords/bounded-function","display_name":"Bounded function","score":0.41970232129096985},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.1300545334815979},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.04956316947937012}],"concepts":[{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.8999511003494263},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7967895269393921},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7694205045700073},{"id":"https://openalex.org/C2779145032","wikidata":"https://www.wikidata.org/wiki/Q191494","display_name":"Digraph","level":2,"score":0.7385308146476746},{"id":"https://openalex.org/C74193536","wikidata":"https://www.wikidata.org/wiki/Q574844","display_name":"Kernel (algebra)","level":2,"score":0.6289010047912598},{"id":"https://openalex.org/C80899671","wikidata":"https://www.wikidata.org/wiki/Q1304193","display_name":"Vertex (graph theory)","level":3,"score":0.43139469623565674},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4202047884464264},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.41970232129096985},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.1300545334815979},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.04956316947937012}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.37236/13653","is_oa":true,"landing_page_url":"https://doi.org/10.37236/13653","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v32i2p53/pdf","source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.37236/13653","is_oa":true,"landing_page_url":"https://doi.org/10.37236/13653","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v32i2p53/pdf","source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"},{"id":"https://openalex.org/F4320323021","display_name":"Nankai University","ror":"https://ror.org/01y1kjr75"},{"id":"https://openalex.org/F4320323993","display_name":"Natural Science Foundation of Tianjin City","ror":null}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4411450266.pdf","grobid_xml":"https://content.openalex.org/works/W4411450266.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W2038088825","https://openalex.org/W1975203667","https://openalex.org/W2370328482","https://openalex.org/W4287882779","https://openalex.org/W2082494488","https://openalex.org/W2740008481","https://openalex.org/W2054349495","https://openalex.org/W2998116189","https://openalex.org/W2081358954","https://openalex.org/W2554461497"],"abstract_inverted_index":{"A":[0,23],"quasi-kernel":[1,45,63],"of":[2,46,112,118,124,141,156],"a":[3,44,62,95,109,116,163],"digraph":[4,42,60],"$D$":[5],"is":[6,106],"an":[7],"independent":[8],"set":[9],"$Q$":[10,17,64],"such":[11,65],"that":[12,39,58,66,71,84,101,148],"every":[13,40,59],"vertex":[14],"can":[15],"reach":[16],"in":[18,131],"at":[19,48,158],"most":[20,49,159],"two":[21],"steps.":[22],"48-year":[24],"conjecture":[25,88,93,105],"made":[26],"by":[27,161],"P.L.":[28],"Erd\u0151s":[29],"and":[30,69],"Sz\u00e9kely,":[31],"known":[32,137],"as":[33],"the":[34,54,75,85,90,102],"small":[35,76,91],"QK":[36,56,77,87,92,104],"conjecture,":[37,57],"says":[38],"sink-free":[41],"contains":[43,61],"size":[47,142,157],"$n/2$.":[50],"Recently,":[51],"Spiro":[52],"posed":[53],"large":[55,86,103],"$|N^-[Q]|\\geq":[67],"n/2$,":[68],"showed":[70],"it":[72],"follows":[73],"from":[74],"conjecture.":[78],"In":[79],"this":[80],"paper,":[81],"we":[82],"establish":[83],"implies":[89],"with":[94,150],"weaker":[96],"constant.":[97],"We":[98,120,146],"also":[99],"show":[100,147],"equivalent":[107],"to":[108,138],"sharp":[110],"version":[111],"it,":[113],"answering":[114],"affirmatively":[115],"question":[117],"Spiro.":[119],"formulate":[121],"variable":[122],"versions":[123],"these":[125],"conjectures,":[126],"which":[127],"are":[128,136],"still":[129],"open":[130],"general.":[132],"Not":[133],"many":[134],"digraphs":[135,149],"have":[139,154],"quasi-kernels":[140,155],"$(1-\\alpha)n$":[143],"or":[144],"less.":[145],"bounded":[151],"dichromatic":[152],"number":[153],"$(1-\\alpha)n$,":[160],"proving":[162],"stronger":[164],"statement.":[165]},"counts_by_year":[],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}
