{"id":"https://openalex.org/W1851714703","doi":"https://doi.org/10.37236/552","title":"Colorings and Nowhere-Zero Flows of Graphs in Terms of Berlekamp's Switching Game","display_name":"Colorings and Nowhere-Zero Flows of Graphs in Terms of Berlekamp's Switching Game","publication_year":2011,"publication_date":"2011-03-24","ids":{"openalex":"https://openalex.org/W1851714703","doi":"https://doi.org/10.37236/552","mag":"1851714703"},"language":"en","primary_location":{"id":"doi:10.37236/552","is_oa":true,"landing_page_url":"https://doi.org/10.37236/552","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v18i1p65/pdf","source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v18i1p65/pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5029447093","display_name":"Uwe Schauz","orcid":"https://orcid.org/0000-0002-3116-9402"},"institutions":[{"id":"https://openalex.org/I134085113","display_name":"King Fahd University of Petroleum and Minerals","ror":"https://ror.org/03yez3163","country_code":"SA","type":"education","lineage":["https://openalex.org/I134085113"]}],"countries":["SA"],"is_corresponding":true,"raw_author_name":"Uwe Schauz","raw_affiliation_strings":["Department of Mathematics and Statistics King Fahd University of Petroleum and Minerals Dhahran 31261, Saudi Arabia","King Fahd University of Petroleum and Minerals,"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Statistics King Fahd University of Petroleum and Minerals Dhahran 31261, Saudi Arabia","institution_ids":["https://openalex.org/I134085113"]},{"raw_affiliation_string":"King Fahd University of Petroleum and Minerals,","institution_ids":["https://openalex.org/I134085113"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5029447093"],"corresponding_institution_ids":["https://openalex.org/I134085113"],"apc_list":null,"apc_paid":null,"fwci":0.5404,"has_fulltext":true,"cited_by_count":8,"citation_normalized_percentile":{"value":0.6869864,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"18","issue":"1","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11797","display_name":"graph theory and CDMA systems","score":0.9997000098228455,"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"}},"topics":[{"id":"https://openalex.org/T11797","display_name":"graph theory and CDMA systems","score":0.9997000098228455,"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"}},{"id":"https://openalex.org/T11476","display_name":"Graph theory and applications","score":0.9983000159263611,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"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":0.9976000189781189,"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/combinatorics","display_name":"Combinatorics","score":0.7928515076637268},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7689696550369263},{"id":"https://openalex.org/keywords/conjecture","display_name":"Conjecture","score":0.5714980363845825},{"id":"https://openalex.org/keywords/vertex","display_name":"Vertex (graph theory)","score":0.48733267188072205},{"id":"https://openalex.org/keywords/zero","display_name":"Zero (linguistics)","score":0.4583335518836975},{"id":"https://openalex.org/keywords/subspace-topology","display_name":"Subspace topology","score":0.45424023270606995},{"id":"https://openalex.org/keywords/order","display_name":"Order (exchange)","score":0.42253419756889343},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.404997318983078},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.39917516708374023}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7928515076637268},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7689696550369263},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.5714980363845825},{"id":"https://openalex.org/C80899671","wikidata":"https://www.wikidata.org/wiki/Q1304193","display_name":"Vertex (graph theory)","level":3,"score":0.48733267188072205},{"id":"https://openalex.org/C2780813799","wikidata":"https://www.wikidata.org/wiki/Q3274237","display_name":"Zero (linguistics)","level":2,"score":0.4583335518836975},{"id":"https://openalex.org/C32834561","wikidata":"https://www.wikidata.org/wiki/Q660730","display_name":"Subspace topology","level":2,"score":0.45424023270606995},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.42253419756889343},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.404997318983078},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.39917516708374023},{"id":"https://openalex.org/C10138342","wikidata":"https://www.wikidata.org/wiki/Q43015","display_name":"Finance","level":1,"score":0.0},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C41895202","wikidata":"https://www.wikidata.org/wiki/Q8162","display_name":"Linguistics","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.37236/552","is_oa":true,"landing_page_url":"https://doi.org/10.37236/552","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v18i1p65/pdf","source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.372.5721","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.372.5721","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.combinatorics.org/ojs/index.php/eljc/article/viewFile/v18i1p65/pdf/","raw_type":"text"}],"best_oa_location":{"id":"doi:10.37236/552","is_oa":true,"landing_page_url":"https://doi.org/10.37236/552","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v18i1p65/pdf","source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320322323","display_name":"King Fahd University of Petroleum and Minerals","ror":"https://ror.org/03yez3163"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W1851714703.pdf","grobid_xml":"https://content.openalex.org/works/W1851714703.grobid-xml"},"referenced_works_count":22,"referenced_works":["https://openalex.org/W3297276","https://openalex.org/W45014588","https://openalex.org/W1497227511","https://openalex.org/W1515228725","https://openalex.org/W1556443961","https://openalex.org/W1566410171","https://openalex.org/W1662383888","https://openalex.org/W1972913697","https://openalex.org/W1976596704","https://openalex.org/W2032637079","https://openalex.org/W2033485192","https://openalex.org/W2051564361","https://openalex.org/W2052664069","https://openalex.org/W2072414205","https://openalex.org/W2134276604","https://openalex.org/W2148360835","https://openalex.org/W2157678969","https://openalex.org/W2324049880","https://openalex.org/W2472744635","https://openalex.org/W2799004609","https://openalex.org/W4285719527","https://openalex.org/W4301890399"],"related_works":["https://openalex.org/W1980381208","https://openalex.org/W2364594919","https://openalex.org/W1861706286","https://openalex.org/W2219338811","https://openalex.org/W2167092671","https://openalex.org/W2149583853","https://openalex.org/W2143002539","https://openalex.org/W2130386332","https://openalex.org/W1120847856","https://openalex.org/W4293472652"],"abstract_inverted_index":{"We":[0],"work":[1],"with":[2,73,153],"a":[3,18,39,43,64,112,128,132,150,160,247],"unifying":[4],"linear":[5],"algebra":[6],"formulation":[7],"for":[8,173,177,198,213,241],"nowhere-zero":[9,40,133,178],"flows":[10,220],"and":[11,15,115,136,157,176,206,215,233,237],"colorings":[12],"of":[13,32,45,82,95,103,142,180,189,210,217,221],"graphs":[14,222],"matrices.":[16,181],"Given":[17],"subspace":[19,61,109],"(code)":[20],"$U\\leq{\\mathbb{Z}_k^n}$":[21],"\u2013":[22,36],"e.g.":[23],"the":[24,27,60,68,86,108,118,139,199],"bond":[25],"or":[26],"cycle":[28],"space":[29,141],"over":[30,164],"${\\mathbb{Z}}_k$":[31,165],"an":[33],"oriented":[34],"graph":[35,129],"we":[37,58,91,192],"call":[38],"tuple":[41],"$f\\in{\\mathbb{Z}_k^n}$":[42],"flow":[44,113],"$U$":[46,62,83,110,121],"if":[47,114,117,135,138,156,159],"$f$":[48],"is":[49,106],"orthogonal":[50],"to":[51,55,80,186],"$U$.":[52],"In":[53,126],"order":[54],"detect":[56],"flows,":[57],"view":[59],"as":[63,224,246],"light":[65,75,119,190],"pattern":[66,120],"on":[67],"$n$-dimensional":[69],"Berlekamp":[70],"Board":[71],"${\\mathbb{Z}_k^n}$":[72],"$k^n$":[74],"bulbs.":[76],"The":[77,100],"lights":[78],"corresponding":[79,162],"elements":[81],"are":[84,88],"ON,":[85],"others":[87],"OFF.":[89],"Then":[90],"allow":[92],"axis-parallel":[93],"switches":[94],"complete":[96],"rows,":[97],"columns,":[98],"etc.":[99],"core":[101],"result":[102],"this":[104],"paper":[105],"that":[107],"has":[111,131,149],"only":[116,137,158],"cannot":[122,144,166],"be":[123,145,167],"switched":[124,146,168],"off.":[125,147,169],"particular,":[127],"$G$":[130,143],"$k$-flow":[134],"${\\mathbb{Z}}_k$-bond":[140],"It":[148],"vertex":[151],"coloring":[152],"$k$":[154],"colors":[155],"certain":[161],"code":[163],"Similar":[170],"statements":[171],"hold":[172],"Tait":[174],"colorings,":[175],"points":[179],"Studying":[182],"different":[183],"normal":[184],"forms":[185],"equivalence":[187],"classes":[188],"patterns,":[191],"find":[193],"various":[194],"new":[195],"equivalents,":[196],"e.g.,":[197],"Four":[200],"Color":[201],"Problem,":[202],"Tutte's":[203],"Flow":[204],"Conjectures":[205],"Jaeger's":[207],"Conjecture.":[208],"Two":[209],"our":[211],"equivalents":[212],"colorability":[214],"existence":[216],"nowhere":[218],"zero":[219],"include":[223],"special":[225],"cases":[226],"results":[227],"by":[228,230,234],"Matiyasevich,":[229],"Bal\u00e1zs":[231],"Szegedy,":[232],"Onn.":[235],"Alon":[236],"Tarsi's":[238],"sufficient":[239],"condition":[240],"$k$-colorability":[242],"also":[243],"arrives,":[244],"remarkably,":[245],"generalized":[248],"full":[249],"equivalent.":[250]},"counts_by_year":[{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":2},{"year":2021,"cited_by_count":1},{"year":2019,"cited_by_count":1},{"year":2018,"cited_by_count":1},{"year":2014,"cited_by_count":2}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}
