{"id":"https://openalex.org/W3083569122","doi":"https://doi.org/10.37236/7862","title":"Chromatic Vertex Folkman Numbers","display_name":"Chromatic Vertex Folkman Numbers","publication_year":2020,"publication_date":"2020-09-04","ids":{"openalex":"https://openalex.org/W3083569122","doi":"https://doi.org/10.37236/7862","mag":"3083569122"},"language":"en","primary_location":{"id":"doi:10.37236/7862","is_oa":true,"landing_page_url":"https://doi.org/10.37236/7862","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v27i3p53/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/v27i3p53/pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5101988903","display_name":"Xiaodong Xu","orcid":"https://orcid.org/0000-0003-3370-8489"},"institutions":[{"id":"https://openalex.org/I4210165264","display_name":"Guangxi Academy of Sciences","ror":"https://ror.org/054x1kd82","country_code":"CN","type":"government","lineage":["https://openalex.org/I4210165264"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Xiaodong Xu","raw_affiliation_strings":["Guangxi Academy of Sciences\r\nNanning, 530007, P.R.China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Guangxi Academy of Sciences\r\nNanning, 530007, P.R.China","institution_ids":["https://openalex.org/I4210165264"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5055172715","display_name":"Meilian Liang","orcid":"https://orcid.org/0000-0002-7839-1174"},"institutions":[{"id":"https://openalex.org/I150807315","display_name":"Guangxi University","ror":"https://ror.org/02c9qn167","country_code":"CN","type":"education","lineage":["https://openalex.org/I150807315"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Meilian Liang","raw_affiliation_strings":["School of Mathematics and Information Science\r\nGuangxi University, Nanning, 530004, P.R.China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematics and Information Science\r\nGuangxi University, Nanning, 530004, P.R.China","institution_ids":["https://openalex.org/I150807315"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5075568055","display_name":"Stanis\u0142aw Radziszowski","orcid":"https://orcid.org/0000-0002-1470-5517"},"institutions":[{"id":"https://openalex.org/I155173764","display_name":"Rochester Institute of Technology","ror":"https://ror.org/00v4yb702","country_code":"US","type":"education","lineage":["https://openalex.org/I155173764"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Stanis\u0142aw P. Radziszowski","raw_affiliation_strings":["Department of Computer Science Rochester Institute of TechnologyRochester, NY"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Computer Science Rochester Institute of TechnologyRochester, NY","institution_ids":["https://openalex.org/I155173764"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":true,"cited_by_count":1,"citation_normalized_percentile":{"value":0.14717836,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"biblio":{"volume":"27","issue":"3","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12541","display_name":"Graph Labeling and Dimension Problems","score":0.9546999931335449,"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/T12541","display_name":"Graph Labeling and Dimension Problems","score":0.9546999931335449,"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/T10374","display_name":"Advanced Graph Theory Research","score":0.9361000061035156,"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/combinatorics","display_name":"Combinatorics","score":0.8933292627334595},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7131402492523193},{"id":"https://openalex.org/keywords/vertex","display_name":"Vertex (graph theory)","score":0.6927505731582642},{"id":"https://openalex.org/keywords/conjecture","display_name":"Conjecture","score":0.6412047147750854},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.6221932768821716},{"id":"https://openalex.org/keywords/chromatic-scale","display_name":"Chromatic scale","score":0.5702235698699951},{"id":"https://openalex.org/keywords/monochromatic-color","display_name":"Monochromatic color","score":0.5683563947677612},{"id":"https://openalex.org/keywords/windmill-graph","display_name":"Windmill graph","score":0.46966302394866943},{"id":"https://openalex.org/keywords/existential-quantification","display_name":"Existential quantification","score":0.4306643009185791},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.41113442182540894},{"id":"https://openalex.org/keywords/graph-power","display_name":"Graph power","score":0.18188408017158508},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.11469367146492004},{"id":"https://openalex.org/keywords/line-graph","display_name":"Line graph","score":0.07027789950370789}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.8933292627334595},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7131402492523193},{"id":"https://openalex.org/C80899671","wikidata":"https://www.wikidata.org/wiki/Q1304193","display_name":"Vertex (graph theory)","level":3,"score":0.6927505731582642},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.6412047147750854},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.6221932768821716},{"id":"https://openalex.org/C196956537","wikidata":"https://www.wikidata.org/wiki/Q202021","display_name":"Chromatic scale","level":2,"score":0.5702235698699951},{"id":"https://openalex.org/C40833965","wikidata":"https://www.wikidata.org/wiki/Q6901438","display_name":"Monochromatic color","level":2,"score":0.5683563947677612},{"id":"https://openalex.org/C73380178","wikidata":"https://www.wikidata.org/wiki/Q8024306","display_name":"Windmill graph","level":5,"score":0.46966302394866943},{"id":"https://openalex.org/C136979486","wikidata":"https://www.wikidata.org/wiki/Q773483","display_name":"Existential quantification","level":2,"score":0.4306643009185791},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.41113442182540894},{"id":"https://openalex.org/C149530733","wikidata":"https://www.wikidata.org/wiki/Q5597091","display_name":"Graph power","level":4,"score":0.18188408017158508},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.11469367146492004},{"id":"https://openalex.org/C203776342","wikidata":"https://www.wikidata.org/wiki/Q1378376","display_name":"Line graph","level":3,"score":0.07027789950370789},{"id":"https://openalex.org/C120665830","wikidata":"https://www.wikidata.org/wiki/Q14620","display_name":"Optics","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.37236/7862","is_oa":true,"landing_page_url":"https://doi.org/10.37236/7862","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v27i3p53/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:10.37236/7862","is_oa":true,"landing_page_url":"https://doi.org/10.37236/7862","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v27i3p53/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":[{"id":"https://openalex.org/G5919171114","display_name":null,"funder_award_id":"2011GXNSFA018142","funder_id":"https://openalex.org/F4320322768","funder_display_name":"Natural Science Foundation of Guangxi Province"}],"funders":[{"id":"https://openalex.org/F4320322768","display_name":"Natural Science Foundation of Guangxi Province","ror":null}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W3083569122.pdf","grobid_xml":"https://content.openalex.org/works/W3083569122.grobid-xml"},"referenced_works_count":15,"referenced_works":["https://openalex.org/W1610839345","https://openalex.org/W1968851262","https://openalex.org/W1984962604","https://openalex.org/W2032331898","https://openalex.org/W2050095135","https://openalex.org/W2068649659","https://openalex.org/W2148550085","https://openalex.org/W2155551829","https://openalex.org/W2155934997","https://openalex.org/W2298378270","https://openalex.org/W2616256309","https://openalex.org/W2795915197","https://openalex.org/W2962872527","https://openalex.org/W2998894191","https://openalex.org/W3128522060"],"related_works":["https://openalex.org/W4302316412","https://openalex.org/W1984480063","https://openalex.org/W2042166619","https://openalex.org/W2033176331","https://openalex.org/W2988746064","https://openalex.org/W2045621845","https://openalex.org/W2059473335","https://openalex.org/W2403055888","https://openalex.org/W2950817880","https://openalex.org/W2091414672"],"abstract_inverted_index":{"For":[0],"graph":[1,67,177],"$G$":[2,37,68,110,143,150,178],"and":[3,20,149,184],"integers":[4],"$a_1":[5],"\\ge":[6,8,10,137,171],"\\cdots":[7],"a_r":[9],"2$,":[11],"we":[12,105,165],"write":[13],"$G":[14,74,85,146],"\\rightarrow":[15,75,86,147],"(a_1":[16,76,87],",\\cdots":[17,51,77,88],",a_r)^v$":[18,89],"if":[19,22,84],"only":[21],"for":[23,38,61,134,168],"every":[24,169],"$r$-coloring":[25],"of":[26,69,121],"the":[27,57,152],"vertex":[28,47],"set":[29],"$V(G)$":[30],"there":[31,63,139,173],"exists":[32,64,174],"a":[33,65,118,175],"monochromatic":[34],"$K_{a_i}$":[35],"in":[36],"some":[39,128],"color":[40],"$i":[41],"\\in":[42],"\\{1,":[43],"\\cdots,":[44],"r\\}$.":[45],"The":[46],"Folkman":[48,108,123],"number":[49,113,156],"$F_v(a_1":[50],",a_r;":[52],"s)$":[53],"is":[54,80],"defined":[55],"as":[56],"smallest":[58,153],"integer":[59],"$n$":[60,71],"which":[62,115],"$K_s$-free":[66],"order":[70],"such":[72,107,144],"that":[73,83,133,145,167],",a_r)^v$.":[78],"It":[79],"well":[81],"known":[82],"then":[90],"$\\chi(G)":[91],"\\geq":[92],"m$,":[93],"where":[94],"$m":[95],"=":[96],"1+":[97],"\\sum_{i=1}^r":[98],"(a_i":[99],"-":[100],"1)$.":[101],"In":[102],"this":[103,161],"paper":[104],"study":[106],"graphs":[109,142],"with":[111,158,182],"chromatic":[112,122,155],"$\\chi(G)=m$,":[114],"leads":[116],"to":[117,160],"new":[119],"concept":[120],"numbers.":[124],"We":[125],"prove":[126],"constructively":[127],"existential":[129],"results,":[130],"among":[131],"others":[132,164],"all":[135],"$r,s":[136],"2$":[138,172],"exist":[140],"$K_{s+1}$-free":[141,176],"(s,\\cdots_r,s)^v$":[148],"has":[151],"possible":[154],"$r(s-1)+1$":[157],"respect":[159],"property.":[162],"Among":[163],"conjecture":[166],"$s":[170],"on":[179],"$F_v(s,s;s+1)$":[180],"vertices":[181],"$\\chi(G)=2s-1$":[183],"$G\\rightarrow":[185],"(s,s)^v$.":[186]},"counts_by_year":[{"year":2018,"cited_by_count":1}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}