{"id":"https://openalex.org/W7214722224","doi":"https://doi.org/10.48550/arxiv.2609.32688","title":"The diameter of recoloring graphs under a maximum average degree bound","display_name":"The diameter of recoloring graphs under a maximum average degree bound","publication_year":2026,"publication_date":"2026-09-26","ids":{"openalex":"https://openalex.org/W7214722224","doi":"https://doi.org/10.48550/arxiv.2609.32688"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.32688","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.32688","pdf_url":"https://arxiv.org/pdf/2609.32688","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":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2609.32688","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5140021046","display_name":"Ruilin Zheng","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I881766915","display_name":"Nanjing University","ror":"https://ror.org/01rxvg760","country_code":"CN","type":"education","lineage":["https://openalex.org/I881766915"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Zheng, Ruilin","raw_affiliation_strings":["School of Mathematics , Nanjing University , Nanjing 210093 , P.R. CHINA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematics , Nanjing University , Nanjing 210093 , P.R. CHINA","institution_ids":["https://openalex.org/I881766915"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5153550925","display_name":"Junying Lu","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I36399199","display_name":"Nanjing University of Science and Technology","ror":"https://ror.org/00xp9wg62","country_code":"CN","type":"education","lineage":["https://openalex.org/I36399199"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Lu, Junying","raw_affiliation_strings":["School of Mathematics and Statistics , Nanjing University of Science and Technology , Nanjing 210094 , P.R. CHINA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematics and Statistics , Nanjing University of Science and Technology , Nanjing 210094 , P.R. CHINA","institution_ids":["https://openalex.org/I36399199"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"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/T10374","display_name":"Advanced Graph Theory Research","score":0.961899995803833,"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":0.961899995803833,"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":0.01940000057220459,"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/T12541","display_name":"Graph Labeling and Dimension Problems","score":0.013199999928474426,"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/graph-diameter","display_name":"graph diameter","score":0.8050000071525574},{"id":"https://openalex.org/keywords/maximum-average-degree","display_name":"maximum average degree","score":0.9860000014305115},{"id":"https://openalex.org/keywords/proper-coloring","display_name":"proper coloring","score":0.777999997138977},{"id":"https://openalex.org/keywords/graph-recoloring","display_name":"graph recoloring","score":0.6909999847412109},{"id":"https://openalex.org/keywords/forests","display_name":"forests","score":0.3540000021457672},{"id":"https://openalex.org/keywords/trees","display_name":"trees","score":0.5889999866485596},{"id":"https://openalex.org/keywords/color-removal","display_name":"color removal","score":0.40400001406669617},{"id":"https://openalex.org/keywords/extremal-graph-theory","display_name":"extremal graph theory","score":0.6299999952316284}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8381999731063843},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.8371000289916992},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.6593000292778015},{"id":"https://openalex.org/C42812","wikidata":"https://www.wikidata.org/wiki/Q1082910","display_name":"Partition (number theory)","level":2,"score":0.5856000185012817},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.5767999887466431},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.46639999747276306},{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.4562000036239624},{"id":"https://openalex.org/C76444178","wikidata":"https://www.wikidata.org/wiki/Q72897900","display_name":"Connectivity","level":3,"score":0.3278000056743622},{"id":"https://openalex.org/C311688","wikidata":"https://www.wikidata.org/wiki/Q2393193","display_name":"Time complexity","level":2,"score":0.289000004529953},{"id":"https://openalex.org/C121134508","wikidata":"https://www.wikidata.org/wiki/Q4949882","display_name":"Bound graph","level":5,"score":0.2572000026702881},{"id":"https://openalex.org/C63553672","wikidata":"https://www.wikidata.org/wiki/Q581168","display_name":"Binary logarithm","level":2,"score":0.2565000057220459},{"id":"https://openalex.org/C88230418","wikidata":"https://www.wikidata.org/wiki/Q131476","display_name":"Graph theory","level":2,"score":0.25600001215934753}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.32688","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.32688","pdf_url":"https://arxiv.org/pdf/2609.32688","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":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2609.32688","doi":"https://doi.org/10.48550/arxiv.2609.32688","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.32688","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":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.32688","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.32688","pdf_url":"https://arxiv.org/pdf/2609.32688","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":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[{"id":"https://openalex.org/G2485548420","display_name":null,"funder_award_id":"12471327","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7214722224.pdf","grobid_xml":"https://content.openalex.org/works/W7214722224.grobid-xml"},"referenced_works_count":1,"referenced_works":["https://openalex.org/W4417032093"],"related_works":["https://openalex.org/W2941238706","https://openalex.org/W4405903447","https://openalex.org/W3083415935","https://openalex.org/W3034428588","https://openalex.org/W7214652332","https://openalex.org/W7129542734","https://openalex.org/W2792834333","https://openalex.org/W4287868079","https://openalex.org/W2005054202","https://openalex.org/W6986766865"],"abstract_inverted_index":{"For":[0],"a":[1,97],"graph":[2,22,63,137],"$G$,":[3],"we":[4,75],"write":[5],"$\\mathrm{mad}(G)$":[6],"for":[7,15,50,133],"its":[8,16],"maximum":[9],"average":[10],"degree":[11],"and":[12,58,102,120,150],"$\\mathrm{diam}":[13,69,142],"G$":[14],"diameter.":[17],"Let":[18],"$R_k(G)$":[19],"be":[20],"the":[21,26,88,113,116,122,128,153,166],"whose":[23],"vertices":[24],"are":[25,37],"proper":[27],"colorings":[28,36],"of":[29,118,156],"$G$":[30,64,138],"with":[31,55,139],"$k$":[32],"colors,":[33],"where":[34,145],"two":[35],"adjacent":[38],"when":[39],"they":[40],"differ":[41],"at":[42,106],"one":[43],"vertex.":[44],"Feghali":[45,92],"(JCTB,":[46],"2021)":[47],"proved":[48,90],"that,":[49],"fixed":[51],"integers":[52],"$d,k\\ge":[53],"1$":[54],"$k\\ge":[56],"d+1$":[57],"every":[59,61,135],"$\\varepsilon>0$,":[60],"$n$-vertex":[62,136],"satisfying":[65],"$\\mathrm{mad}(G)\\le":[66],"d-\\varepsilon$":[67],"has":[68],"R_k(G)=O_{d,k,\\varepsilon}(n(\\log":[70],"n)^{d-1})$.":[71],"In":[72],"this":[73],"article,":[74],"prove":[76],"that":[77],"\\[":[78],"\\mathrm{diam}":[79],"R_k(G)=O_{d,k,\\varepsilon}\\!\\left(":[80],"n(\\log":[81],"n)^{\\left\\lfloor":[82],"(d-1)/(k-d)\\right\\rfloor}":[83],"\\right),":[84],"\\]":[85],"which":[86],"extends":[87],"result":[89],"by":[91],"directly.":[93],"The":[94],"proof":[95],"uses":[96],"partition":[98],"into":[99],"independent":[100],"layers":[101,119],"removes":[103],"$k-d$":[104],"colors":[105],"each":[107],"recursive":[108],"stage.":[109],"We":[110],"also":[111],"improve":[112],"bound":[114],"on":[115,162],"number":[117],"determine":[121],"best":[123,170],"possible":[124],"linear":[125],"coefficient":[126,167],"in":[127],"forest":[129],"case.":[130],"More":[131],"precisely,":[132],"$0<\\varepsilon<2$,":[134],"$\\mathrm{mad}(G)\\le2-\\varepsilon$":[140],"satisfies":[141],"R_3(G)\\le\u03c1_M":[143],"n$,":[144],"$M=\\lfloor2/\\varepsilon\\rfloor$,":[146],"$\u03c1_M=\\max_{1\\le":[147],"m\\le":[148],"M}D_m/m$,":[149],"$D_m$":[151],"is":[152,169],"largest":[154],"diameter":[155],"$R_3(T)$":[157],"over":[158],"all":[159],"trees":[160],"$T$":[161],"$m$":[163],"vertices.":[164],"Moreover,":[165],"$\u03c1_M$":[168],"possible.":[171]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-30T00:00:00"}