{"id":"https://openalex.org/W2044145837","doi":"https://doi.org/10.1137/s0097539798338175","title":"On Approximately Counting Colorings of Small Degree Graphs","display_name":"On Approximately Counting Colorings of Small Degree Graphs","publication_year":1999,"publication_date":"1999-01-01","ids":{"openalex":"https://openalex.org/W2044145837","doi":"https://doi.org/10.1137/s0097539798338175","mag":"2044145837"},"language":"en","primary_location":{"id":"doi:10.1137/s0097539798338175","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s0097539798338175","pdf_url":null,"source":{"id":"https://openalex.org/S153560523","display_name":"SIAM Journal on Computing","issn_l":"0097-5397","issn":["0097-5397","1095-7111"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Computing","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5069943660","display_name":"Russ Bubley","orcid":null},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Russ Bubley","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5019297501","display_name":"Martin Dyer","orcid":"https://orcid.org/0000-0002-2018-0374"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Martin Dyer","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5031194096","display_name":"Catherine Greenhill","orcid":"https://orcid.org/0000-0001-6998-2282"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Catherine Greenhill","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5090122841","display_name":"Mark Jerrum","orcid":"https://orcid.org/0000-0003-0863-7279"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Mark Jerrum","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5069943660"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":2.7849,"has_fulltext":false,"cited_by_count":45,"citation_normalized_percentile":{"value":0.91206407,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"29","issue":"2","first_page":"387","last_page":"400"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12056","display_name":"Markov Chains and Monte Carlo Methods","score":1.0,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"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/T12056","display_name":"Markov Chains and Monte Carlo Methods","score":1.0,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"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/T11152","display_name":"Stochastic processes and statistical mechanics","score":0.9995999932289124,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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/T10588","display_name":"Mathematical Dynamics and Fractals","score":0.9840999841690063,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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/combinatorics","display_name":"Combinatorics","score":0.8227187395095825},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7595453262329102},{"id":"https://openalex.org/keywords/markov-chain","display_name":"Markov chain","score":0.6707769632339478},{"id":"https://openalex.org/keywords/degree","display_name":"Degree (music)","score":0.6066855192184448},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.5460749268531799},{"id":"https://openalex.org/keywords/vertex","display_name":"Vertex (graph theory)","score":0.5005037784576416},{"id":"https://openalex.org/keywords/counting-problem","display_name":"Counting problem","score":0.45550015568733215},{"id":"https://openalex.org/keywords/chromatic-polynomial","display_name":"Chromatic polynomial","score":0.4327290654182434},{"id":"https://openalex.org/keywords/mixing","display_name":"Mixing (physics)","score":0.42007142305374146},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.39817309379577637}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.8227187395095825},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7595453262329102},{"id":"https://openalex.org/C98763669","wikidata":"https://www.wikidata.org/wiki/Q176645","display_name":"Markov chain","level":2,"score":0.6707769632339478},{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.6066855192184448},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5460749268531799},{"id":"https://openalex.org/C80899671","wikidata":"https://www.wikidata.org/wiki/Q1304193","display_name":"Vertex (graph theory)","level":3,"score":0.5005037784576416},{"id":"https://openalex.org/C16592021","wikidata":"https://www.wikidata.org/wiki/Q5177154","display_name":"Counting problem","level":2,"score":0.45550015568733215},{"id":"https://openalex.org/C126385604","wikidata":"https://www.wikidata.org/wiki/Q1087609","display_name":"Chromatic polynomial","level":3,"score":0.4327290654182434},{"id":"https://openalex.org/C138777275","wikidata":"https://www.wikidata.org/wiki/Q6884054","display_name":"Mixing (physics)","level":2,"score":0.42007142305374146},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.39817309379577637},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C24890656","wikidata":"https://www.wikidata.org/wiki/Q82811","display_name":"Acoustics","level":1,"score":0.0},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/s0097539798338175","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s0097539798338175","pdf_url":null,"source":{"id":"https://openalex.org/S153560523","display_name":"SIAM Journal on Computing","issn_l":"0097-5397","issn":["0097-5397","1095-7111"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Computing","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":19,"referenced_works":["https://openalex.org/W1573043775","https://openalex.org/W1607251184","https://openalex.org/W1986979529","https://openalex.org/W2012032973","https://openalex.org/W2028283969","https://openalex.org/W2039763834","https://openalex.org/W2049712767","https://openalex.org/W2055119378","https://openalex.org/W2063342497","https://openalex.org/W2106285343","https://openalex.org/W2109260999","https://openalex.org/W2140422174","https://openalex.org/W2140572013","https://openalex.org/W2147632348","https://openalex.org/W2148360835","https://openalex.org/W2168256531","https://openalex.org/W3111890340","https://openalex.org/W4233358567","https://openalex.org/W4301890399"],"related_works":["https://openalex.org/W2356950793","https://openalex.org/W2017638499","https://openalex.org/W2033753932","https://openalex.org/W2128225235","https://openalex.org/W2167159596","https://openalex.org/W4302403014","https://openalex.org/W2953164658","https://openalex.org/W4211170257","https://openalex.org/W3102254024","https://openalex.org/W2005253487"],"abstract_inverted_index":{"We":[0,58,108,142],"consider":[1],"approximate":[2,122],"counting":[3,123,133,152],"of":[4,6,40,48,54,75,78,93,105,114,150,155,158],"colorings":[5,33,101],"an":[7,110],"n-vertex":[8],"graph":[9,32,50],"using":[10,102],"rapidly":[11],"mixing":[12,69,92,120],"Markov":[13,98],"chains.":[14],"It":[15],"has":[16],"been":[17],"shown":[18],"by":[19,22,51,70],"Jerrum":[20],"and":[21,24,170],"Salas":[23],"Sokal":[25],"that":[26,60,131,147],"a":[27,52,64,85,94,144],"simple":[28],"random":[29],"walk":[30],"on":[31,100],"would":[34],"mix":[35],"rapidly,":[36],"provided":[37],"the":[38,44,49,72,103,148,153],"number":[39,154],"colors":[41],"k":[42],"exceeded":[43],"maximum":[45,79,161],"degree":[46,80,162],"$\\Delta$":[47,163],"factor":[53],"at":[55],"least":[56],"2.":[57],"prove":[59],"this":[61],"is":[62,134,164],"not":[63],"necessary":[65],"condition":[66],"for":[67],"rapid":[68,91,119],"considering":[71],"simplest":[73],"case":[74],"5-coloring":[76],"graphs":[77,159],"3.":[81],"Our":[82],"proof":[83,87,146],"involves":[84],"computer-assisted":[86],"technique":[88],"to":[89,112,136],"establish":[90],"new":[95],"\"heat":[96],"bath\"":[97],"chain":[99],"method":[104],"path":[106],"coupling.":[107],"outline":[109],"extension":[111],"7-colorings":[113],"triangle-free":[115],"4-regular":[116],"graphs.":[117],"Since":[118],"implies":[121],"in":[124,129],"polynomial":[125,140],"time,":[126],"we":[127],"show":[128],"contrast":[130],"exact":[132],"unlikely":[135],"be":[137],"possible":[138],"(in":[139],"time).":[141],"give":[143],"general":[145],"problem":[149],"exactly":[151],"proper":[156],"k-colorings":[157],"with":[160],"$#":[165],"P$-complete":[166],"whenever":[167],"$k\\geq":[168],"3$":[169],"$\\Delta":[171],"\\geq":[172],"3$.":[173]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2022,"cited_by_count":1},{"year":2020,"cited_by_count":3},{"year":2019,"cited_by_count":2},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":4},{"year":2016,"cited_by_count":3},{"year":2015,"cited_by_count":1},{"year":2013,"cited_by_count":2},{"year":2012,"cited_by_count":4}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}