{"id":"https://openalex.org/W2029630058","doi":"https://doi.org/10.1017/s0963548303005947","title":"On the Number of Edges in Random Planar Graphs","display_name":"On the Number of Edges in Random Planar Graphs","publication_year":2004,"publication_date":"2004-03-01","ids":{"openalex":"https://openalex.org/W2029630058","doi":"https://doi.org/10.1017/s0963548303005947","mag":"2029630058"},"language":"en","primary_location":{"id":"doi:10.1017/s0963548303005947","is_oa":false,"landing_page_url":"https://doi.org/10.1017/s0963548303005947","pdf_url":null,"source":{"id":"https://openalex.org/S7291079","display_name":"Combinatorics Probability Computing","issn_l":"0963-5483","issn":["0963-5483","1469-2163"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310311721","host_organization_name":"Cambridge University Press","host_organization_lineage":["https://openalex.org/P4310311721","https://openalex.org/P4310311702"],"host_organization_lineage_names":["Cambridge University Press","University of Cambridge"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Combinatorics, Probability and 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/A5047303000","display_name":"Stefanie Gerke","orcid":"https://orcid.org/0000-0002-9426-1708"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"STEFANIE GERKE","raw_affiliation_strings":["TU M\u00fcnchen, Institut f\u00fcr Informatik, Arcisstra\u00dfe 21, 80290 M\u00fcnchen, Germany (e-mail:","TU M\u00fcnchen, Institut f\u00fcr Informatik, Arcisstra\u00dfe 21, 80290 M\u00fcnchen, Germany (e-mail: [email\u00a0protected])#TAB#"],"affiliations":[{"raw_affiliation_string":"TU M\u00fcnchen, Institut f\u00fcr Informatik, Arcisstra\u00dfe 21, 80290 M\u00fcnchen, Germany (e-mail:","institution_ids":[]},{"raw_affiliation_string":"TU M\u00fcnchen, Institut f\u00fcr Informatik, Arcisstra\u00dfe 21, 80290 M\u00fcnchen, Germany (e-mail: [email\u00a0protected])#TAB#","institution_ids":[]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5111437501","display_name":"Colin McDiarmid","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":false,"raw_author_name":"COLIN McDIARMID","raw_affiliation_strings":["University of Oxford, Department of Statistics, South Parks Road, Oxford OX1 3TG, UK (e-mail:","University of Oxford, Department of Statistics, South Parks Road, Oxford OX1 3TG, UK (e-mail: cmcd@stats.ox.ac.uk)"],"affiliations":[{"raw_affiliation_string":"University of Oxford, Department of Statistics, South Parks Road, Oxford OX1 3TG, UK (e-mail:","institution_ids":["https://openalex.org/I40120149"]},{"raw_affiliation_string":"University of Oxford, Department of Statistics, South Parks Road, Oxford OX1 3TG, UK (e-mail: cmcd@stats.ox.ac.uk)","institution_ids":["https://openalex.org/I40120149"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5047303000"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":6.7193,"has_fulltext":false,"cited_by_count":54,"citation_normalized_percentile":{"value":0.97846801,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":97},"biblio":{"volume":"13","issue":"2","first_page":"165","last_page":"183"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11152","display_name":"Stochastic processes and statistical mechanics","score":0.9667999744415283,"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"}},"topics":[{"id":"https://openalex.org/T11152","display_name":"Stochastic processes and statistical mechanics","score":0.9667999744415283,"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/T10996","display_name":"Computational Geometry and Mesh Generation","score":0.9391000270843506,"subfield":{"id":"https://openalex.org/subfields/1704","display_name":"Computer Graphics and Computer-Aided Design"},"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/T10374","display_name":"Advanced Graph Theory Research","score":0.9038000106811523,"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":0.7971296310424805},{"id":"https://openalex.org/keywords/planar-graph","display_name":"Planar graph","score":0.7327069044113159},{"id":"https://openalex.org/keywords/planar","display_name":"Planar","score":0.693665087223053},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6451550722122192},{"id":"https://openalex.org/keywords/random-graph","display_name":"Random graph","score":0.553633987903595},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.5461264252662659},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.47383007407188416},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.42794108390808105},{"id":"https://openalex.org/keywords/planar-straight-line-graph","display_name":"Planar straight-line graph","score":0.4104394018650055},{"id":"https://openalex.org/keywords/1-planar-graph","display_name":"1-planar graph","score":0.2666760981082916},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.14987578988075256},{"id":"https://openalex.org/keywords/line-graph","display_name":"Line graph","score":0.11709043383598328},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.06320559978485107}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7971296310424805},{"id":"https://openalex.org/C101837359","wikidata":"https://www.wikidata.org/wiki/Q547823","display_name":"Planar graph","level":3,"score":0.7327069044113159},{"id":"https://openalex.org/C134786449","wikidata":"https://www.wikidata.org/wiki/Q3391255","display_name":"Planar","level":2,"score":0.693665087223053},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6451550722122192},{"id":"https://openalex.org/C47458327","wikidata":"https://www.wikidata.org/wiki/Q910404","display_name":"Random graph","level":3,"score":0.553633987903595},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.5461264252662659},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.47383007407188416},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.42794108390808105},{"id":"https://openalex.org/C131930043","wikidata":"https://www.wikidata.org/wiki/Q7200964","display_name":"Planar straight-line graph","level":5,"score":0.4104394018650055},{"id":"https://openalex.org/C102192266","wikidata":"https://www.wikidata.org/wiki/Q4545823","display_name":"1-planar graph","level":4,"score":0.2666760981082916},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.14987578988075256},{"id":"https://openalex.org/C203776342","wikidata":"https://www.wikidata.org/wiki/Q1378376","display_name":"Line graph","level":3,"score":0.11709043383598328},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.06320559978485107},{"id":"https://openalex.org/C121684516","wikidata":"https://www.wikidata.org/wiki/Q7600677","display_name":"Computer graphics (images)","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1017/s0963548303005947","is_oa":false,"landing_page_url":"https://doi.org/10.1017/s0963548303005947","pdf_url":null,"source":{"id":"https://openalex.org/S7291079","display_name":"Combinatorics Probability Computing","issn_l":"0963-5483","issn":["0963-5483","1469-2163"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310311721","host_organization_name":"Cambridge University Press","host_organization_lineage":["https://openalex.org/P4310311721","https://openalex.org/P4310311702"],"host_organization_lineage_names":["Cambridge University Press","University of Cambridge"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Combinatorics, Probability and Computing","raw_type":"journal-article"},{"id":"pmh:oai:ora.ox.ac.uk:uuid:8334dc9e-efa7-44fa-ba43-b28d7da5a42a","is_oa":false,"landing_page_url":"https://ora.ox.ac.uk/objects/uuid:8334dc9e-efa7-44fa-ba43-b28d7da5a42a","pdf_url":null,"source":{"id":"https://openalex.org/S4306402636","display_name":"Oxford University Research Archive (ORA) (University of Oxford)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I40120149","host_organization_name":"University of Oxford","host_organization_lineage":["https://openalex.org/I40120149"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symplectic Elements at Oxford","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":3,"referenced_works":["https://openalex.org/W1974936141","https://openalex.org/W2054600724","https://openalex.org/W2134147092"],"related_works":["https://openalex.org/W1491152805","https://openalex.org/W2081273953","https://openalex.org/W3002061974","https://openalex.org/W165147621","https://openalex.org/W4291736767","https://openalex.org/W2951173809","https://openalex.org/W2009558719","https://openalex.org/W2002953601","https://openalex.org/W2990956612","https://openalex.org/W3216105633"],"abstract_inverted_index":{"We":[0],"consider":[1],"random":[2,21],"planar":[3,57],"graphs":[4],"on":[5,43,59],"n":[6,60],"labelled":[7],"nodes,":[8],"and":[9,62,69],"show":[10],"in":[11,70],"particular":[12,71],"that":[13,51,74],"if":[14,75],"the":[15,23,44,47],"graph":[16,58],"is":[17,28,77],"picked":[18],"uniformly":[19],"at":[20,29,78,89],"then":[22,88],"expected":[24],"number":[25,92],"of":[26,46,49,93],"edges":[27,50,64,94],"least":[30,90],"13/7n":[31,80],"+":[32],"o(n).":[33],"To":[34],"prove":[35],"this":[36,91],"result":[37],"we":[38,72],"give":[39],"a":[40,56,84],"lower":[41],"bound":[42],"size":[45],"set":[48],"can":[52,95],"be":[53,96],"added":[54],"to":[55],"nodes":[61],"m":[63,76],"while":[65],"keeping":[66],"it":[67],"planar,":[68],"see":[73],"most":[79],"-":[81],"c":[82],"(for":[83],"suitable":[85],"constant":[86],"c)":[87],"added.":[97]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":3},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":2},{"year":2019,"cited_by_count":1},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":2},{"year":2013,"cited_by_count":1}],"updated_date":"2026-04-04T16:13:02.066488","created_date":"2025-10-10T00:00:00"}
