{"id":"https://openalex.org/W7213406924","doi":"https://doi.org/10.13140/rg.2.2.31675.58407","title":"SHARP RAINBOW PATH COVERS IN DENSE AND COMPLETE MULTIPARTITE GRAPHS","display_name":"SHARP RAINBOW PATH COVERS IN DENSE AND COMPLETE MULTIPARTITE GRAPHS","publication_year":2026,"publication_date":"2026-09-16","ids":{"openalex":"https://openalex.org/W7213406924","doi":"https://doi.org/10.13140/rg.2.2.31675.58407"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.18740","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.18740","pdf_url":"https://arxiv.org/pdf/2609.18740","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/2609.18740","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5100698272","display_name":"Xiao\u2010Chuan Liu","orcid":"https://orcid.org/0000-0002-6079-7707","observed_orcids":["https://orcid.org/0000-0002-6079-7707"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Xiao-Chuan Liu","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5151877986","display_name":"Boyan Xu","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Boyan Xu","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5151884852","display_name":"X U Yang","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"X U Yang","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"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/T11329","display_name":"Limits and Structures in Graph Theory","score":0.9980000257492065,"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"}},"topics":[{"id":"https://openalex.org/T11329","display_name":"Limits and Structures in Graph Theory","score":0.9980000257492065,"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/T10374","display_name":"Advanced Graph Theory Research","score":0.00139999995008111,"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/T12541","display_name":"Graph Labeling and Dimension Problems","score":0.0003000000142492354,"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/rainbow-paths","display_name":"rainbow paths","score":0.8840000033378601},{"id":"https://openalex.org/keywords/proper-edge-coloring","display_name":"proper edge coloring","score":0.7820000052452087},{"id":"https://openalex.org/keywords/complete-multipartite-graphs","display_name":"complete multipartite graphs","score":0.906000018119812},{"id":"https://openalex.org/keywords/dense-graphs","display_name":"dense graphs","score":0.9440000057220459},{"id":"https://openalex.org/keywords/minimum-degree","display_name":"minimum degree","score":0.6269999742507935},{"id":"https://openalex.org/keywords/graph-decomposition","display_name":"graph decomposition","score":0.8149999976158142}],"concepts":[{"id":"https://openalex.org/C166733315","wikidata":"https://www.wikidata.org/wiki/Q6934831","display_name":"Multipartite","level":4,"score":0.5942000150680542},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5728999972343445},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5507000088691711},{"id":"https://openalex.org/C2777735758","wikidata":"https://www.wikidata.org/wiki/Q817765","display_name":"Path (computing)","level":2,"score":0.5364000201225281},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.38909998536109924},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.28630000352859497},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.2768999934196472},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.27559998631477356},{"id":"https://openalex.org/C184720557","wikidata":"https://www.wikidata.org/wiki/Q7825049","display_name":"Topology (electrical circuits)","level":2,"score":0.27469998598098755},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.2632000148296356},{"id":"https://openalex.org/C88230418","wikidata":"https://www.wikidata.org/wiki/Q131476","display_name":"Graph theory","level":2,"score":0.25290000438690186}],"mesh":[],"locations_count":3,"locations":[{"id":"pmh:oai:arXiv.org:2609.18740","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.18740","pdf_url":"https://arxiv.org/pdf/2609.18740","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.2609.18740","doi":"https://doi.org/10.48550/arxiv.2609.18740","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.18740","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"},{"id":"doi:10.13140/rg.2.2.31675.58407","doi":"https://doi.org/10.13140/rg.2.2.31675.58407","is_oa":false,"landing_page_url":"https://doi.org/10.13140/rg.2.2.31675.58407","pdf_url":null,"source":null,"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"article-journal"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.18740","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.18740","pdf_url":"https://arxiv.org/pdf/2609.18740","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":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7213406924.pdf","grobid_xml":"https://content.openalex.org/works/W7213406924.grobid-xml"},"referenced_works_count":1,"referenced_works":["https://openalex.org/W7201895230"],"related_works":["https://openalex.org/W2559261669","https://openalex.org/W2098679506","https://openalex.org/W4307308541","https://openalex.org/W659604157","https://openalex.org/W1620856143","https://openalex.org/W36146487","https://openalex.org/W1810953614","https://openalex.org/W2067931229","https://openalex.org/W1968948734","https://openalex.org/W4248704047"],"abstract_inverted_index":{"A":[0],"path":[1],"in":[2],"a":[3,17,22,130,147],"properly":[4,60],"edge-colored":[5,61],"graph":[6,23,63],"is":[7,76],"rainbow":[8,32,126],"if":[9],"its":[10],"edges":[11],"have":[12],"pairwise":[13],"distinct":[14],"colors.":[15],"For":[16],"proper":[18,48],"edge-coloring":[19],"$c$":[20],"of":[21,31,44,50,109,114,124],"$G$,":[24],"let":[25,39],"$\\operatorname{rpc}(G,c)$":[26,45],"be":[27,41],"the":[28,42,73,110,115],"minimum":[29,65],"number":[30,111],"paths":[33],"needed":[34],"to":[35],"cover":[36],"$E(G)$,":[37],"and":[38,94,96,101,112,135,142,146],"$\\operatorname{rpc}(G)$":[40,82],"maximum":[43],"over":[46,106],"all":[47,107],"edge-colorings":[49],"$G$.":[51],"We":[52,79],"prove":[53],"that,":[54],"for":[55,84,138,151],"every":[56,59,85],"fixed":[57],"$0<\u03b1<1$,":[58],"$n$-vertex":[62],"with":[64,129,143],"degree":[66],"at":[67],"least":[68],"$\u03b1n$":[69],"satisfies":[70],"$\\operatorname{rpc}(G,c)\\leq(1+o(1))n/2$,":[71],"where":[72],"coefficient":[74],"$1/2$":[75],"best":[77],"possible.":[78],"also":[80],"determine":[81],"asymptotically":[83],"complete":[86,152],"multipartite":[87,153],"graph.":[88],"If":[89],"$G=K_{n_1,\\ldots,n_r}$":[90],"has":[91],"order":[92],"$n$":[93],"largest":[95],"smallest":[97],"part":[98],"sizes":[99,113],"$M$":[100],"$s$,":[102],"respectively,":[103],"then,":[104],"uniformly":[105],"choices":[108],"parts,":[116],"$\\operatorname{rpc}(G)=(1+o(1))\\max\\{\\min\\{\\lfloor":[117],"n/2\\rfloor,n-M\\},(n-s)/2\\}$.":[118],"The":[119],"proof":[120],"combines":[121],"pseudorandom":[122],"packings":[123],"globally":[125],"linear":[127],"forests":[128],"decomposition":[131],"into":[132],"dense":[133,140],"parts":[134],"prescribed":[136],"avoidance":[137],"arbitrary":[139],"graphs,":[141],"reserved":[144],"connectors":[145],"direct":[148],"dominant-part":[149],"argument":[150],"graphs.":[154]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-17T00:00:00"}