{"id":"https://openalex.org/W2767152628","doi":"https://doi.org/10.37236/7163","title":"Strengthening $(a,b)$-Choosability Results to $(a,b)$-Paintability","display_name":"Strengthening $(a,b)$-Choosability Results to $(a,b)$-Paintability","publication_year":2017,"publication_date":"2017-10-20","ids":{"openalex":"https://openalex.org/W2767152628","doi":"https://doi.org/10.37236/7163","mag":"2767152628"},"language":"en","primary_location":{"id":"doi:10.37236/7163","is_oa":true,"landing_page_url":"https://doi.org/10.37236/7163","pdf_url":null,"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://doi.org/10.37236/7163","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5073136526","display_name":"Thomas A. Mahoney","orcid":"https://orcid.org/0000-0002-7401-8727"},"institutions":[{"id":"https://openalex.org/I93131583","display_name":"Emporia State University","ror":"https://ror.org/04e6r1478","country_code":"US","type":"education","lineage":["https://openalex.org/I93131583"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Thomas Mahoney","raw_affiliation_strings":["Emporia State University"],"affiliations":[{"raw_affiliation_string":"Emporia State University","institution_ids":["https://openalex.org/I93131583"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5073136526"],"corresponding_institution_ids":["https://openalex.org/I93131583"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.18508745,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"24","issue":"4","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12288","display_name":"Optimization and Search Problems","score":0.8051999807357788,"subfield":{"id":"https://openalex.org/subfields/1705","display_name":"Computer Networks and Communications"},"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/T12288","display_name":"Optimization and Search Problems","score":0.8051999807357788,"subfield":{"id":"https://openalex.org/subfields/1705","display_name":"Computer Networks and Communications"},"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":0.7839000225067139,"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":0.8953083753585815},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8167923092842102},{"id":"https://openalex.org/keywords/vertex","display_name":"Vertex (graph theory)","score":0.7049415111541748},{"id":"https://openalex.org/keywords/conjecture","display_name":"Conjecture","score":0.7009642124176025},{"id":"https://openalex.org/keywords/disjoint-sets","display_name":"Disjoint sets","score":0.6307544112205505},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.5957902669906616},{"id":"https://openalex.org/keywords/list-coloring","display_name":"List coloring","score":0.49582013487815857},{"id":"https://openalex.org/keywords/graph-power","display_name":"Graph power","score":0.0993196964263916}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.8953083753585815},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8167923092842102},{"id":"https://openalex.org/C80899671","wikidata":"https://www.wikidata.org/wiki/Q1304193","display_name":"Vertex (graph theory)","level":3,"score":0.7049415111541748},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.7009642124176025},{"id":"https://openalex.org/C45340560","wikidata":"https://www.wikidata.org/wiki/Q215382","display_name":"Disjoint sets","level":2,"score":0.6307544112205505},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.5957902669906616},{"id":"https://openalex.org/C199594403","wikidata":"https://www.wikidata.org/wiki/Q1130404","display_name":"List coloring","level":5,"score":0.49582013487815857},{"id":"https://openalex.org/C149530733","wikidata":"https://www.wikidata.org/wiki/Q5597091","display_name":"Graph power","level":4,"score":0.0993196964263916},{"id":"https://openalex.org/C203776342","wikidata":"https://www.wikidata.org/wiki/Q1378376","display_name":"Line graph","level":3,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.37236/7163","is_oa":true,"landing_page_url":"https://doi.org/10.37236/7163","pdf_url":null,"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:10.37236/7163","is_oa":true,"landing_page_url":"https://doi.org/10.37236/7163","pdf_url":null,"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":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W4256429076","https://openalex.org/W1971174658","https://openalex.org/W2099195351","https://openalex.org/W2348092930","https://openalex.org/W2012063272","https://openalex.org/W2138685729","https://openalex.org/W2062850566","https://openalex.org/W4308716484","https://openalex.org/W2098284475","https://openalex.org/W26693340"],"abstract_inverted_index":{"Let":[0],"$a,b\\in\\mathbb{N}$.":[1],"A":[2],"graph":[3,208,217],"$G$":[4,83,147,151,183,187,209,221],"is":[5,64,84,91,99,129,134,137,148,152,184,188,222],"$(a,b)$-choosable":[6],"if":[7,86,146],"for":[8,40,154,177,205],"any":[9,41,155,178,206,219],"list":[10],"assignment":[11],"$L$":[12],"such":[13,33,114],"that":[14,34,79,115,124,145,182,204],"$|L(v)|\\ge":[15],"a$,":[16],"there":[17,98],"exists":[18],"a":[19,27,54,61,104],"coloring":[20,68,105],"in":[21,106],"which":[22,107,136],"each":[23,52,88,108],"vertex":[24,89,109],"$v$":[25,90,110],"receives":[26,111],"set":[28,55],"$C(v)$":[29],"of":[30,48,56,74,121,168,173,192,197],"$b$":[31,112],"colors":[32,113],"$C(v)\\subseteq":[35],"L(v)$":[36],"and":[37,66,170,172,194,196,200,218],"$C(u)\\cap":[38],"C(w)=\\emptyset$":[39],"$uw\\in":[42],"E(G)$.":[43],"In":[44,141],"the":[45,67],"online":[46],"version":[47],"this":[49,164],"problem,":[50],"on":[51],"round,":[53],"vertices":[57,76,117],"allowed":[58,92],"to":[59,77,93,102],"receive":[60,78,118],"particular":[62],"color":[63],"marked,":[65],"algorithm":[69,101],"chooses":[70],"an":[71,100,212],"independent":[72],"subset":[73],"these":[75],"color.":[80],"We":[81],"say":[82],"$(a,b)$-paintable":[85,130],"when":[87,132,138,186],"be":[94],"marked":[95],"$a$":[96],"times,":[97],"produce":[103],"adjacent":[116],"disjoint":[119],"sets":[120],"colors.We":[122],"show":[123],"every":[125],"odd":[126,213],"cycle":[127,214],"$C_{2k+1}$":[128],"exactly":[131],"it":[133],"$(a,b)$-chosable,":[135],"$a\\ge2b+\\lceil":[139],"b/k\\rceil$.":[140],"2009,":[142],"Zhu":[143],"conjectured":[144],"$(a,1)$-paintable,":[149],"then":[150],"$(am,m)$-paintable":[153],"$m\\in\\mathbb{N}$.":[156],"The":[157],"following":[158],"results":[159,167],"make":[160],"partial":[161],"progress":[162],"towards":[163],"conjecture.":[165],"Strengthening":[166,190],"Tuza":[169,193],"Voigt,":[171,195],"Schauz,":[174,201],"we":[175,202],"prove":[176,203],"$m":[179],"\\in":[180],"\\mathbb{N}$":[181],"$(5m,m)$-paintable":[185],"planar.":[189],"work":[191],"Hladky,":[198],"Kral,":[199],"connected":[207],"other":[210],"than":[211],"or":[215],"complete":[216],"$m\\in\\mathbb{N}$,":[220],"$(\\Delta(G)m,m)$-paintable.":[223]},"counts_by_year":[],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}
