{"id":"https://openalex.org/W7130349401","doi":"https://doi.org/10.4230/lipics.csl.2026.2","title":"The Logic Behind Colour Refinement (Invited Talk)","display_name":"The Logic Behind Colour Refinement (Invited Talk)","publication_year":2026,"publication_date":"2026-01-01","ids":{"openalex":"https://openalex.org/W7130349401","doi":"https://doi.org/10.4230/lipics.csl.2026.2"},"language":"en","primary_location":{"id":"pmh:oai:drops-oai.dagstuhl.de:25426","is_oa":true,"landing_page_url":"https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2026.2","pdf_url":null,"source":{"id":"https://openalex.org/S4306402524","display_name":"Leibniz-Zentrum f\u00fcr Informatik (Schloss Dagstuhl)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I2799853480","host_organization_name":"Schloss Dagstuhl \u2013 Leibniz Center for Informatics","host_organization_lineage":["https://openalex.org/I2799853480"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"publishedVersion"},"type":"article","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2026.2","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5045167968","display_name":"Sandra Kiefer","orcid":null},"institutions":[{"id":"https://openalex.org/I40120149","display_name":"University of Oxford","ror":"https://ror.org/052gg0110","country_code":"GB","type":"education","lineage":["https://openalex.org/I40120149"]}],"countries":["GB"],"is_corresponding":true,"raw_author_name":"Kiefer, Sandra","raw_affiliation_strings":["University of Oxford, UK"],"raw_orcid":"https://orcid.org/0000-0003-4614-9444","affiliations":[{"raw_affiliation_string":"University of Oxford, UK","institution_ids":["https://openalex.org/I40120149"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5045167968"],"corresponding_institution_ids":["https://openalex.org/I40120149"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.38288057,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"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":0.4607999920845032,"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":0.4607999920845032,"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/T12029","display_name":"DNA and Biological Computing","score":0.0544000007212162,"subfield":{"id":"https://openalex.org/subfields/1312","display_name":"Molecular Biology"},"field":{"id":"https://openalex.org/fields/13","display_name":"Biochemistry, Genetics and Molecular Biology"},"domain":{"id":"https://openalex.org/domains/1","display_name":"Life Sciences"}},{"id":"https://openalex.org/T12541","display_name":"Graph Labeling and Dimension Problems","score":0.044599998742341995,"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/multiset","display_name":"Multiset","score":0.7210999727249146},{"id":"https://openalex.org/keywords/vertex","display_name":"Vertex (graph theory)","score":0.6385999917984009},{"id":"https://openalex.org/keywords/partition","display_name":"Partition (number theory)","score":0.43209999799728394},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.3894999921321869},{"id":"https://openalex.org/keywords/automorphism","display_name":"Automorphism","score":0.3336000144481659},{"id":"https://openalex.org/keywords/data-structure","display_name":"Data structure","score":0.3203999996185303},{"id":"https://openalex.org/keywords/robustness","display_name":"Robustness (evolution)","score":0.3165999948978424},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.3107999861240387}],"concepts":[{"id":"https://openalex.org/C2779623528","wikidata":"https://www.wikidata.org/wiki/Q864377","display_name":"Multiset","level":2,"score":0.7210999727249146},{"id":"https://openalex.org/C80899671","wikidata":"https://www.wikidata.org/wiki/Q1304193","display_name":"Vertex (graph theory)","level":3,"score":0.6385999917984009},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5701000094413757},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.4433000087738037},{"id":"https://openalex.org/C42812","wikidata":"https://www.wikidata.org/wiki/Q1082910","display_name":"Partition (number theory)","level":2,"score":0.43209999799728394},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.3894999921321869},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.37290000915527344},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3546999990940094},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3467000126838684},{"id":"https://openalex.org/C118712358","wikidata":"https://www.wikidata.org/wiki/Q782566","display_name":"Automorphism","level":2,"score":0.3336000144481659},{"id":"https://openalex.org/C162319229","wikidata":"https://www.wikidata.org/wiki/Q175263","display_name":"Data structure","level":2,"score":0.3203999996185303},{"id":"https://openalex.org/C63479239","wikidata":"https://www.wikidata.org/wiki/Q7353546","display_name":"Robustness (evolution)","level":3,"score":0.3165999948978424},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.3107999861240387},{"id":"https://openalex.org/C311688","wikidata":"https://www.wikidata.org/wiki/Q2393193","display_name":"Time complexity","level":2,"score":0.3046000003814697},{"id":"https://openalex.org/C134537474","wikidata":"https://www.wikidata.org/wiki/Q17144832","display_name":"Naturalness","level":2,"score":0.2802000045776367},{"id":"https://openalex.org/C9376300","wikidata":"https://www.wikidata.org/wiki/Q168817","display_name":"Algebraic number","level":2,"score":0.273499995470047},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.26649999618530273},{"id":"https://openalex.org/C61665672","wikidata":"https://www.wikidata.org/wiki/Q303100","display_name":"Graph isomorphism","level":4,"score":0.263700008392334},{"id":"https://openalex.org/C136979486","wikidata":"https://www.wikidata.org/wiki/Q773483","display_name":"Existential quantification","level":2,"score":0.26179999113082886},{"id":"https://openalex.org/C148043351","wikidata":"https://www.wikidata.org/wiki/Q4456944","display_name":"Current (fluid)","level":2,"score":0.2590999901294708},{"id":"https://openalex.org/C3018263672","wikidata":"https://www.wikidata.org/wiki/Q1296251","display_name":"Efficient algorithm","level":2,"score":0.2581000030040741},{"id":"https://openalex.org/C88230418","wikidata":"https://www.wikidata.org/wiki/Q131476","display_name":"Graph theory","level":2,"score":0.2524000108242035}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:drops-oai.dagstuhl.de:25426","is_oa":true,"landing_page_url":"https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2026.2","pdf_url":null,"source":{"id":"https://openalex.org/S4306402524","display_name":"Leibniz-Zentrum f\u00fcr Informatik (Schloss Dagstuhl)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I2799853480","host_organization_name":"Schloss Dagstuhl \u2013 Leibniz Center for Informatics","host_organization_lineage":["https://openalex.org/I2799853480"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"publishedVersion"},{"id":"doi:10.4230/lipics.csl.2026.2","is_oa":true,"landing_page_url":"https://doi.org/10.4230/lipics.csl.2026.2","pdf_url":null,"source":{"id":"https://openalex.org/S7407052059","display_name":"Dagstuhl Research Online Publication Server","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":""}],"best_oa_location":{"id":"pmh:oai:drops-oai.dagstuhl.de:25426","is_oa":true,"landing_page_url":"https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2026.2","pdf_url":null,"source":{"id":"https://openalex.org/S4306402524","display_name":"Leibniz-Zentrum f\u00fcr Informatik (Schloss Dagstuhl)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I2799853480","host_organization_name":"Schloss Dagstuhl \u2013 Leibniz Center for Informatics","host_organization_lineage":["https://openalex.org/I2799853480"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"publishedVersion"},"sustainable_development_goals":[{"score":0.4028148651123047,"display_name":"Sustainable cities and communities","id":"https://metadata.un.org/sdg/11"}],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"Colour":[0,177,238,272,330,414,476],"Refinement":[1,178,239,331,415,477],"is":[2,41,85,111,171,194,248,293,299,370,520],"a":[3,8,39,62,106,161,187,195,236,249,281,285,300,314,345,352,381,423,443,469,492,495,506],"combinatorial":[4,116],"algorithm":[5,24,77,89,110,279],"that":[6,100,126,206,210,251,439,485],"computes":[7,280],"vertex":[9,40,67,224,357],"colouring":[10,28,84,283],"for":[11,123,214,284],"an":[12,407],"input":[13],"graph":[14,188,286,444],"to":[15,98,118,130,144,174,183,257,267,320,334,340,396,409,465,501],"reveal":[16],"its":[17,44,53,154,310],"structural":[18],"properties.":[19],"Each":[20],"iteration":[21],"of":[22,38,50,65,134,138,153,164,202,217,264,271,316,329,344,354,373,389,426,442,462,472,483,494,508],"the":[23,26,35,48,66,73,76,82,88,109,113,131,135,139,167,172,198,203,211,218,223,260,268,274,318,321,327,341,349,359,371,376,387,392,410,413,418,440,459,523],"refines":[25],"current":[27,45],"by":[29,43],"assessing":[30],"local":[31],"information.":[32],"More":[33],"precisely,":[34],"new":[36],"colour":[37,46,70],"determined":[42],"and":[47,156,158,190,244,289,309,391,417,431,505,526,533,541],"multiset":[49],"colours":[51,182,225,337],"in":[52,93,147,186,197,295,302,402,445],"neighbourhood.":[54],"This":[55,254,278],"refinement":[56],"procedure":[57,416],"continues":[58],"until":[59],"it":[60,160],"reaches":[61],"stable":[63],"partition":[64],"set":[68],"into":[69,228],"classes.":[71],"On":[72],"practical":[74],"side,":[75],"admits":[78],"fast":[79],"implementations.":[80],"Because":[81],"final":[83,181],"isomorphism-invariant,":[86],"executing":[87],"on":[90,385,429,432,458,522],"two":[91,184,219,241,399],"graphs":[92,242,430,484,504,511],"parallel":[94],"can":[95,447],"be":[96,448],"used":[97],"demonstrate":[99],"they":[101],"are":[102,307],"not":[103,128],"isomorphic.":[104],"From":[105],"theoretical":[107,498],"perspective,":[108],"arguably":[112],"most":[114,168,365],"straightforward":[115],"approach":[117,500],"detecting":[119],"asymmetries":[120],"-":[121],"specifically":[122],"distinguishing":[124,346],"vertices":[125,185,401],"do":[127],"belong":[129],"same":[132],"orbit":[133],"automorphism":[136],"group":[137],"graph.":[140],"Its":[141],"numerous":[142],"connections":[143,170],"other":[145],"areas":[146],"computer":[148],"science":[149],"stand":[150],"as":[151,380],"evidence":[152],"robustness":[155],"naturalness":[157],"make":[159],"fascinating":[162],"object":[163],"research.":[165],"Among":[166],"elegant":[169],"link":[173,319,411],"counting":[175],"logic.":[176],"assigns":[179],"distinct":[180,336],"if":[189,192,243,246,288,291],"only":[191,245,290,305],"there":[193,247,298],"formula":[196,212],"two-variable":[199],"fragment":[200,262],"C\u00b2":[201,446],"logic":[204,322,419],"C":[205,265,323],"distinguishes":[207,240,252],"them,":[208],"meaning":[209],"holds":[213],"precisely":[215],"one":[216,232],"vertices.":[220,374,467],"In":[221],"fact,":[222,317],"translate":[226],"directly":[227],"logical":[229],"formulas":[230],"with":[231,491,512],"free":[233],"variable.":[234],"As":[235,313],"consequence,":[237],"C\u00b2-sentence":[250],"them.":[253],"correspondence":[255],"extends":[256],"higher":[258],"dimensions:":[259],"k-variable":[261],"C^k":[263,303],"corresponds":[266,338],"(k-1)-dimensional":[269,275],"extension":[270],"Refinement,":[273],"Weisfeiler-Leman":[276],"algorithm.":[277],"unique":[282],"G":[287,292,308],"definable":[294],"C^k,":[296],"i.e.":[297],"sentence":[301],"whose":[304],"models":[306],"isomorphic":[311],"copies.":[312],"matter":[315],"goes":[324],"even":[325],"deeper:":[326],"number":[328,372,388],"iterations":[332,350,390],"required":[333,395,464],"compute":[335],"exactly":[339],"quantifier":[342,393,460],"depth":[343,394,461],"formula.":[347],"Since":[348],"induce":[351],"sequence":[353],"strictly":[355],"nested":[356],"partitions,":[358],"process":[360],"must":[361],"terminate":[362],"after":[363],"at":[364],"n-1":[366,378],"rounds,":[367],"where":[368],"n":[369],"Consequently,":[375],"value":[377],"serves":[379],"trivial":[382],"upper":[383],"bound":[384],"both":[386],"distinguish":[397,466],"any":[398],"(distinguishable)":[400],"C\u00b2.":[403,420],"My":[404],"talk":[405,519],"provides":[406],"introduction":[408],"between":[412],"We":[421,452,489],"revisit":[422],"simple":[424],"characterisation":[425,437],"their":[427],"expressivity":[428],"general":[433],"relational":[434],"structures.":[435],"The":[436,518],"implies":[438],"definability":[441],"checked":[449],"very":[450],"efficiently.":[451],"then":[453],"discuss":[454],"tight":[455],"lower":[456],"bounds":[457],"C\u00b2-formulas":[463],"Through":[468],"thorough":[470],"analysis":[471],"computational":[473],"data":[474],"from":[475],"executions,":[478],"we":[479],"constructed":[480],"infinite":[481],"families":[482],"witness":[486],"those":[487],"bounds.":[488],"finish":[490],"presentation":[493],"recent":[496],"purely":[497],"reverse-engineering":[499],"finding":[502],"long-refinement":[503],"classification":[507],"all":[509],"such":[510],"small":[513],"(or,":[514],"equivalently,":[515],"large)":[516],"degrees.":[517],"based":[521],"collaborations[Sandra":[524],"Kiefer":[525,532,537],"T.":[527],"Devini":[528],"de":[529],"Mel,":[530],"2026;":[531],"McKay,":[534],"2020;":[535],"Sandra":[536],"et":[538],"al.,":[539],"2022]":[540],"unpublished":[542],"work.":[543]},"counts_by_year":[],"updated_date":"2026-02-19T06:31:58.851227","created_date":"2026-02-19T00:00:00"}