{"id":"https://openalex.org/W2076335141","doi":"https://doi.org/10.1002/(sici)1097-0118(199901)30:1<7::aid-jgt2>3.0.co;2-4","title":"Proof of a conjecture of Alan Hartman","display_name":"Proof of a conjecture of Alan Hartman","publication_year":1999,"publication_date":"1999-01-01","ids":{"openalex":"https://openalex.org/W2076335141","doi":"https://doi.org/10.1002/(sici)1097-0118(199901)30:1<7::aid-jgt2>3.0.co;2-4","mag":"2076335141"},"language":"en","primary_location":{"id":"doi:10.1002/(sici)1097-0118(199901)30:1<7::aid-jgt2>3.0.co;2-4","is_oa":false,"landing_page_url":"https://doi.org/10.1002/(sici)1097-0118(199901)30:1<7::aid-jgt2>3.0.co;2-4","pdf_url":null,"source":{"id":"https://openalex.org/S193368155","display_name":"Journal of Graph Theory","issn_l":"0364-9024","issn":["0364-9024","1097-0118"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320595","host_organization_name":"Wiley","host_organization_lineage":["https://openalex.org/P4310320595"],"host_organization_lineage_names":["Wiley"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Graph Theory","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5090790432","display_name":"Qizhang Liu","orcid":"https://orcid.org/0009-0005-6027-7110"},"institutions":[{"id":"https://openalex.org/I165932596","display_name":"National University of Singapore","ror":"https://ror.org/01tgyzw49","country_code":"SG","type":"education","lineage":["https://openalex.org/I165932596"]}],"countries":["SG"],"is_corresponding":true,"raw_author_name":"Q. Z. Liu","raw_affiliation_strings":["National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260","National University of Singapore, 10 Kent Ridge Crescent, Singapore?119260"],"affiliations":[{"raw_affiliation_string":"National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260","institution_ids":["https://openalex.org/I165932596"]},{"raw_affiliation_string":"National University of Singapore, 10 Kent Ridge Crescent, Singapore?119260","institution_ids":["https://openalex.org/I165932596"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5013588304","display_name":"Hian Poh Yap","orcid":null},"institutions":[{"id":"https://openalex.org/I165932596","display_name":"National University of Singapore","ror":"https://ror.org/01tgyzw49","country_code":"SG","type":"education","lineage":["https://openalex.org/I165932596"]}],"countries":["SG"],"is_corresponding":false,"raw_author_name":"H. P. Yap","raw_affiliation_strings":["National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260","National University of Singapore, 10 Kent Ridge Crescent, Singapore?119260"],"affiliations":[{"raw_affiliation_string":"National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260","institution_ids":["https://openalex.org/I165932596"]},{"raw_affiliation_string":"National University of Singapore, 10 Kent Ridge Crescent, Singapore?119260","institution_ids":["https://openalex.org/I165932596"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5090790432"],"corresponding_institution_ids":["https://openalex.org/I165932596"],"apc_list":{"value":3760,"currency":"USD","value_usd":3760},"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.20303892,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"30","issue":"1","first_page":"7","last_page":"17"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12541","display_name":"Graph Labeling and Dimension Problems","score":0.9997000098228455,"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":0.9997000098228455,"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":0.9994000196456909,"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/T11797","display_name":"graph theory and CDMA systems","score":0.9976999759674072,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.8514183163642883},{"id":"https://openalex.org/keywords/conjecture","display_name":"Conjecture","score":0.7745843529701233},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7555871605873108},{"id":"https://openalex.org/keywords/vertex","display_name":"Vertex (graph theory)","score":0.5515555739402771},{"id":"https://openalex.org/keywords/disjoint-sets","display_name":"Disjoint sets","score":0.5004494190216064},{"id":"https://openalex.org/keywords/star","display_name":"Star (game theory)","score":0.48192667961120605},{"id":"https://openalex.org/keywords/tree","display_name":"Tree (set theory)","score":0.458662211894989},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.43599390983581543},{"id":"https://openalex.org/keywords/spanning-tree","display_name":"Spanning tree","score":0.4178019165992737},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3949306607246399}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.8514183163642883},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.7745843529701233},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7555871605873108},{"id":"https://openalex.org/C80899671","wikidata":"https://www.wikidata.org/wiki/Q1304193","display_name":"Vertex (graph theory)","level":3,"score":0.5515555739402771},{"id":"https://openalex.org/C45340560","wikidata":"https://www.wikidata.org/wiki/Q215382","display_name":"Disjoint sets","level":2,"score":0.5004494190216064},{"id":"https://openalex.org/C2780897414","wikidata":"https://www.wikidata.org/wiki/Q7600592","display_name":"Star (game theory)","level":2,"score":0.48192667961120605},{"id":"https://openalex.org/C113174947","wikidata":"https://www.wikidata.org/wiki/Q2859736","display_name":"Tree (set theory)","level":2,"score":0.458662211894989},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.43599390983581543},{"id":"https://openalex.org/C64331007","wikidata":"https://www.wikidata.org/wiki/Q831672","display_name":"Spanning tree","level":2,"score":0.4178019165992737},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3949306607246399},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1002/(sici)1097-0118(199901)30:1<7::aid-jgt2>3.0.co;2-4","is_oa":false,"landing_page_url":"https://doi.org/10.1002/(sici)1097-0118(199901)30:1<7::aid-jgt2>3.0.co;2-4","pdf_url":null,"source":{"id":"https://openalex.org/S193368155","display_name":"Journal of Graph Theory","issn_l":"0364-9024","issn":["0364-9024","1097-0118"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320595","host_organization_name":"Wiley","host_organization_lineage":["https://openalex.org/P4310320595"],"host_organization_lineage_names":["Wiley"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Graph Theory","raw_type":"journal-article"},{"id":"doi:10.1002/(sici)1097-0118(199901)30:1<7::aid-jgt2>3.3.co;2-w","is_oa":false,"landing_page_url":"https://doi.org/10.1002/(sici)1097-0118(199901)30:1<7::aid-jgt2>3.3.co;2-w","pdf_url":null,"source":{"id":"https://openalex.org/S193368155","display_name":"Journal of Graph Theory","issn_l":"0364-9024","issn":["0364-9024","1097-0118"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320595","host_organization_name":"Wiley","host_organization_lineage":["https://openalex.org/P4310320595"],"host_organization_lineage_names":["Wiley"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Graph Theory","raw_type":"journal-article"},{"id":"pmh:oai:scholarbank.nus.edu.sg:10635/45078","is_oa":false,"landing_page_url":"http://scholarbank.nus.edu.sg/handle/10635/45078","pdf_url":null,"source":{"id":"https://openalex.org/S7407052290","display_name":"National University of Singapore","issn_l":null,"issn":[],"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Scopus","raw_type":"Article"},{"id":"mag:2076335141","is_oa":false,"landing_page_url":"https://dblp.uni-trier.de/db/journals/jgt/jgt30.html#LiuY99","pdf_url":null,"source":{"id":"https://openalex.org/S193368155","display_name":"Journal of Graph Theory","issn_l":"0364-9024","issn":["0364-9024","1097-0118"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320595","host_organization_name":"Wiley","host_organization_lineage":["https://openalex.org/P4310320595"],"host_organization_lineage_names":["Wiley"],"type":"journal"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":"Journal of Graph Theory","raw_type":null}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":5,"referenced_works":["https://openalex.org/W1964924588","https://openalex.org/W1980409271","https://openalex.org/W2093209566","https://openalex.org/W4213214460","https://openalex.org/W4299363239"],"related_works":["https://openalex.org/W2109511316","https://openalex.org/W3106196710","https://openalex.org/W2987876456","https://openalex.org/W2883232345","https://openalex.org/W2887840724","https://openalex.org/W3158169011","https://openalex.org/W2772485695","https://openalex.org/W1708213524","https://openalex.org/W2371508910","https://openalex.org/W2962784526","https://openalex.org/W3200785092","https://openalex.org/W1645173339","https://openalex.org/W3197202779","https://openalex.org/W2768594358","https://openalex.org/W2134453562","https://openalex.org/W1972212864","https://openalex.org/W3006967677","https://openalex.org/W2745779757","https://openalex.org/W2276153108","https://openalex.org/W3045988463"],"abstract_inverted_index":{"A":[0,34],"tree":[1,35,58],"T":[2,36,59,80,88],"is":[3,10,37,41,89],"said":[4],"to":[5,27],"be":[6],"bad,":[7],"if":[8,39,84,87],"it":[9,40],"the":[11,21,24,31,77],"vertex-disjoint":[12],"union":[13],"of":[14,23,30,51,60,72,79],"two":[15],"stars":[16],"plus":[17],"an":[18,28],"edge":[19],"joining":[20],"center":[22],"first":[25],"star":[26],"end-vertex":[29],"second":[32],"star.":[33],"good,":[38],"not":[42],"bad.":[43],"In":[44],"this":[45],"article,":[46],"we":[47],"prove":[48],"a":[49,68],"conjecture":[50],"Alan":[52],"Hartman":[53],"that,":[54],"for":[55],"any":[56],"spanning":[57],"K2m,":[61],"where":[62],"m":[63],"\u2265":[64],"4,":[65],"there":[66],"exists":[67],"(2m":[69],"\u2212":[70],"1)-edge-coloring":[71],"K2m":[73],"such":[74],"that":[75],"all":[76],"edges":[78],"receive":[81],"distinct":[82],"colors":[83],"and":[85],"only":[86],"good.":[90],"\u00a9":[91],"1999":[92,103],"John":[93],"Wiley":[94],"&":[95],"Sons,":[96],"Inc.":[97],"J":[98],"Graph":[99],"Theory":[100],"30:":[101],"7\u201317,":[102]},"counts_by_year":[],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}