{"id":"https://openalex.org/W7212028587","doi":"https://doi.org/10.48550/arxiv.2609.06706","title":"Rainbow paths in directed graphs","display_name":"Rainbow paths in directed graphs","publication_year":2026,"publication_date":"2026-09-06","ids":{"openalex":"https://openalex.org/W7212028587","doi":"https://doi.org/10.48550/arxiv.2609.06706"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.06706","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.06706","pdf_url":"https://arxiv.org/pdf/2609.06706","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/2609.06706","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5112916674","display_name":"Chase Wilson","orcid":"https://orcid.org/0009-0008-1005-8640","observed_orcids":["https://orcid.org/0009-0008-1005-8640"]},"institutions":[{"id":"https://openalex.org/I36258959","display_name":"University of California San Diego","ror":"https://ror.org/0168r3w48","country_code":"US","type":"education","lineage":["https://openalex.org/I36258959"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Wilson, Chase","raw_affiliation_strings":["Department of Mathematics , University of California , San Diego , CA , 92093-0112 USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics , University of California , San Diego , CA , 92093-0112 USA","institution_ids":["https://openalex.org/I36258959"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5112916674"],"corresponding_institution_ids":["https://openalex.org/I36258959"],"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":0.9961000084877014,"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":0.9961000084877014,"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":0.0013000000035390258,"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/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.0007999999797903001,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/rainbow-paths","display_name":"rainbow paths","score":0.9039999842643738},{"id":"https://openalex.org/keywords/directed-graphs","display_name":"directed graphs","score":0.8399999737739563},{"id":"https://openalex.org/keywords/combinatorial-group-theory","display_name":"combinatorial group theory","score":0.7799999713897705},{"id":"https://openalex.org/keywords/cayley-graphs","display_name":"Cayley graphs","score":0.7720000147819519},{"id":"https://openalex.org/keywords/asymptotic-results","display_name":"asymptotic results","score":0.7710000276565552},{"id":"https://openalex.org/keywords/d-regular-graphs","display_name":"d-regular graphs","score":0.6159999966621399},{"id":"https://openalex.org/keywords/partial-products","display_name":"partial products","score":0.6729999780654907}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7214000225067139},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6753000020980835},{"id":"https://openalex.org/C2777735758","wikidata":"https://www.wikidata.org/wiki/Q817765","display_name":"Path (computing)","level":2,"score":0.5461000204086304},{"id":"https://openalex.org/C2776558979","wikidata":"https://www.wikidata.org/wiki/Q1052","display_name":"Rainbow","level":2,"score":0.5386000275611877},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5159000158309937},{"id":"https://openalex.org/C146380142","wikidata":"https://www.wikidata.org/wiki/Q1137726","display_name":"Directed graph","level":2,"score":0.49630001187324524},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.48590001463890076},{"id":"https://openalex.org/C136979486","wikidata":"https://www.wikidata.org/wiki/Q773483","display_name":"Existential quantification","level":2,"score":0.3824000060558319},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.3822000026702881},{"id":"https://openalex.org/C120204988","wikidata":"https://www.wikidata.org/wiki/Q859174","display_name":"Cayley graph","level":3,"score":0.3723999857902527},{"id":"https://openalex.org/C74197172","wikidata":"https://www.wikidata.org/wiki/Q1195339","display_name":"Directed acyclic graph","level":2,"score":0.34769999980926514},{"id":"https://openalex.org/C2778355321","wikidata":"https://www.wikidata.org/wiki/Q17079427","display_name":"Identity (music)","level":2,"score":0.33379998803138733},{"id":"https://openalex.org/C88230418","wikidata":"https://www.wikidata.org/wiki/Q131476","display_name":"Graph theory","level":2,"score":0.32710000872612},{"id":"https://openalex.org/C2781311116","wikidata":"https://www.wikidata.org/wiki/Q83306","display_name":"Group (periodic table)","level":2,"score":0.3174999952316284},{"id":"https://openalex.org/C2778307483","wikidata":"https://www.wikidata.org/wiki/Q5149038","display_name":"Colored","level":2,"score":0.2904999852180481},{"id":"https://openalex.org/C200288055","wikidata":"https://www.wikidata.org/wiki/Q2621792","display_name":"Element (criminal law)","level":2,"score":0.2775999903678894},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.27720001339912415},{"id":"https://openalex.org/C1465435","wikidata":"https://www.wikidata.org/wiki/Q2916352","display_name":"Longest path problem","level":4,"score":0.2766000032424927},{"id":"https://openalex.org/C122818955","wikidata":"https://www.wikidata.org/wiki/Q1060343","display_name":"Independent set","level":3,"score":0.2606000006198883},{"id":"https://openalex.org/C38754835","wikidata":"https://www.wikidata.org/wiki/Q2003238","display_name":"Strongly connected component","level":2,"score":0.25529998540878296}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.06706","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.06706","pdf_url":"https://arxiv.org/pdf/2609.06706","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.2609.06706","doi":"https://doi.org/10.48550/arxiv.2609.06706","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.06706","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.06706","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.06706","pdf_url":"https://arxiv.org/pdf/2609.06706","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":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7212028587.pdf","grobid_xml":"https://content.openalex.org/works/W7212028587.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W7133527338","https://openalex.org/W7077896492","https://openalex.org/W2145217822","https://openalex.org/W2060414214","https://openalex.org/W2913825130","https://openalex.org/W1484490151","https://openalex.org/W2071066571","https://openalex.org/W7096865986","https://openalex.org/W177428913","https://openalex.org/W2094363710"],"abstract_inverted_index":{"An":[0],"old":[1],"problem":[2,134],"in":[3,138],"combinatorial":[4],"group":[5,10],"theory":[6],"asks,":[7],"given":[8,184],"a":[9,13,102,144,153,172],"$\u0393$":[11,93,206],"and":[12,94,128,190],"subset":[14,75,96],"$S":[15,97],"\\subseteq":[16,98,106],"\u0393$,":[17,99],"when":[18],"does":[19,150],"there":[20,100,151],"exist":[21,152],"an":[22,51,85,163],"ordering":[23,52],"$s_1,":[24],"\\cdots,":[25],"s_k$":[26],"of":[27,30,88,123,135,156,166,175,198,205],"the":[28,34,66,78,121,132,192,202],"elements":[29],"$S$":[31,56],"such":[32,50,108],"that":[33,70,109,169],"partial":[35],"products":[36],"$\\prod_{i":[37],"=":[38,111],"1}^j":[39],"s_i$,":[40],"$1":[41],"\\leq":[42,44],"j":[43],"k$,":[45],"are":[46],"all":[47],"distinct.":[48],"If":[49,141],"exists,":[53],"we":[54,118],"call":[55],"rearrangeable.":[57,82],"There":[58],"have":[59],"been":[60],"many":[61],"conjectures":[62],"about":[63],"rearrangeable":[64,103,196],"subsets,":[65],"most":[67],"general":[68],"being":[69],"for":[71],"every":[72,74],"group,":[73],"not":[76],"containing":[77],"identity":[79],"element":[80],"is":[81,143],"We":[83,161],"prove":[84],"asymptotic":[86,164],"version":[87,165],"this:":[89],"For":[90],"any":[91,95],"Group":[92],"exists":[101],"set":[104,211],"$S'":[105],"S$":[107],"$|S'|":[110],"|S|":[112],"-":[113,159,178],"o(|S|)$.":[114],"To":[115],"do":[116],"this":[117],"build":[119],"upon":[120],"work":[122],"Buci\u0107,":[124,186],"Frederickson,":[125,187],"M\u00fcyesser,":[126],"Pokrovskiy,":[127],"Yepremyan":[129],"focusing":[130],"on":[131,195],"following":[133],"independent":[136],"interest":[137],"Graph":[139],"Theory.":[140],"$G$":[142,170],"d-regular":[145],"properly":[146],"colored":[147],"directed":[148],"graph":[149,204],"rainbow":[154,173],"path":[155,174],"length":[157,176],"$d":[158,177],"1$?":[160],"establish":[162],"this,":[167],"proving":[168],"contains":[171],"o(d)$.":[179],"This":[180],"solves":[181],"two":[182],"problems":[183],"by":[185,200],"et":[188],"al.":[189],"proves":[191],"above":[193],"result":[194],"subsets":[197],"groups":[199],"considering":[201],"Cayley":[203],"with":[207],"(not":[208],"necessarily":[209],"generating)":[210],"$S$.":[212]},"counts_by_year":[],"updated_date":"2026-11-06T12:05:46.655746","created_date":"2026-09-10T00:00:00"}