{"id":"https://openalex.org/W2956003067","doi":"https://doi.org/10.37236/8847","title":"A Lower Bound on the Average Degree Forcing a Minor","display_name":"A Lower Bound on the Average Degree Forcing a Minor","publication_year":2020,"publication_date":"2020-04-03","ids":{"openalex":"https://openalex.org/W2956003067","doi":"https://doi.org/10.37236/8847","mag":"2956003067"},"language":"en","primary_location":{"id":"doi:10.37236/8847","is_oa":true,"landing_page_url":"https://doi.org/10.37236/8847","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v27i2p4/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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v27i2p4/pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5031639997","display_name":"Sergey Norin","orcid":"https://orcid.org/0000-0003-4833-7983"},"institutions":[{"id":"https://openalex.org/I5023651","display_name":"McGill University","ror":"https://ror.org/01pxwe438","country_code":"CA","type":"education","lineage":["https://openalex.org/I5023651"]}],"countries":["CA"],"is_corresponding":false,"raw_author_name":"Sergey Norin","raw_affiliation_strings":["McGill University"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"McGill University","institution_ids":["https://openalex.org/I5023651"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5108103165","display_name":"Bruce Reed","orcid":"https://orcid.org/0009-0002-5656-8941"},"institutions":[{"id":"https://openalex.org/I241749","display_name":"University of Cambridge","ror":"https://ror.org/013meh722","country_code":"GB","type":"education","lineage":["https://openalex.org/I241749"]},{"id":"https://openalex.org/I5023651","display_name":"McGill University","ror":"https://ror.org/01pxwe438","country_code":"CA","type":"education","lineage":["https://openalex.org/I5023651"]},{"id":"https://openalex.org/I56590836","display_name":"Monash University","ror":"https://ror.org/02bfwt286","country_code":"AU","type":"education","lineage":["https://openalex.org/I56590836"]}],"countries":["AU","CA","GB"],"is_corresponding":false,"raw_author_name":"Bruce Reed","raw_affiliation_strings":["McGill University Montr\u00e9al, Canada","Department of Pure Mathematics and Mathematical Statistics University of Cambridge United Kingdom","School of Mathematics Monash University Melbourne, Australia","Department of Mathematics and Statistics","School of Computer Science McGill University Montr\u00e9al, Canada"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"McGill University Montr\u00e9al, Canada","institution_ids":["https://openalex.org/I5023651"]},{"raw_affiliation_string":"Department of Pure Mathematics and Mathematical Statistics University of Cambridge United Kingdom","institution_ids":["https://openalex.org/I241749"]},{"raw_affiliation_string":"School of Mathematics Monash University Melbourne, Australia","institution_ids":["https://openalex.org/I56590836"]},{"raw_affiliation_string":"Department of Mathematics and Statistics","institution_ids":[]},{"raw_affiliation_string":"School of Computer Science McGill University Montr\u00e9al, Canada","institution_ids":["https://openalex.org/I5023651"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5084708925","display_name":"Andrew Thomason","orcid":"https://orcid.org/0000-0002-1805-4221"},"institutions":[{"id":"https://openalex.org/I241749","display_name":"University of Cambridge","ror":"https://ror.org/013meh722","country_code":"GB","type":"education","lineage":["https://openalex.org/I241749"]},{"id":"https://openalex.org/I5023651","display_name":"McGill University","ror":"https://ror.org/01pxwe438","country_code":"CA","type":"education","lineage":["https://openalex.org/I5023651"]},{"id":"https://openalex.org/I56590836","display_name":"Monash University","ror":"https://ror.org/02bfwt286","country_code":"AU","type":"education","lineage":["https://openalex.org/I56590836"]}],"countries":["AU","CA","GB"],"is_corresponding":false,"raw_author_name":"Andrew Thomason","raw_affiliation_strings":["School of Mathematics Monash University Melbourne, Australia","McGill University Montr\u00e9al, Canada","School of Computer Science McGill University Montr\u00e9al, Canada","Department of Pure Mathematics and Mathematical Statistics University of Cambridge United Kingdom","Department of Mathematics and Statistics"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematics Monash University Melbourne, Australia","institution_ids":["https://openalex.org/I56590836"]},{"raw_affiliation_string":"McGill University Montr\u00e9al, Canada","institution_ids":["https://openalex.org/I5023651"]},{"raw_affiliation_string":"School of Computer Science McGill University Montr\u00e9al, Canada","institution_ids":["https://openalex.org/I5023651"]},{"raw_affiliation_string":"Department of Pure Mathematics and Mathematical Statistics University of Cambridge United Kingdom","institution_ids":["https://openalex.org/I241749"]},{"raw_affiliation_string":"Department of Mathematics and Statistics","institution_ids":[]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5066302647","display_name":"David R. Wood","orcid":"https://orcid.org/0000-0001-8866-3041"},"institutions":[{"id":"https://openalex.org/I241749","display_name":"University of Cambridge","ror":"https://ror.org/013meh722","country_code":"GB","type":"education","lineage":["https://openalex.org/I241749"]},{"id":"https://openalex.org/I5023651","display_name":"McGill University","ror":"https://ror.org/01pxwe438","country_code":"CA","type":"education","lineage":["https://openalex.org/I5023651"]},{"id":"https://openalex.org/I56590836","display_name":"Monash University","ror":"https://ror.org/02bfwt286","country_code":"AU","type":"education","lineage":["https://openalex.org/I56590836"]}],"countries":["AU","CA","GB"],"is_corresponding":false,"raw_author_name":"David R. Wood","raw_affiliation_strings":["School of Computer Science McGill University Montr\u00e9al, Canada","Department of Pure Mathematics and Mathematical Statistics University of Cambridge United Kingdom","McGill University Montr\u00e9al, Canada","Department of Mathematics and Statistics","School of Mathematics Monash University Melbourne, Australia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Computer Science McGill University Montr\u00e9al, Canada","institution_ids":["https://openalex.org/I5023651"]},{"raw_affiliation_string":"Department of Pure Mathematics and Mathematical Statistics University of Cambridge United Kingdom","institution_ids":["https://openalex.org/I241749"]},{"raw_affiliation_string":"McGill University Montr\u00e9al, Canada","institution_ids":["https://openalex.org/I5023651"]},{"raw_affiliation_string":"Department of Mathematics and Statistics","institution_ids":[]},{"raw_affiliation_string":"School of Mathematics Monash University Melbourne, Australia","institution_ids":["https://openalex.org/I56590836"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":4,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":2.7378,"has_fulltext":true,"cited_by_count":7,"citation_normalized_percentile":{"value":0.88711166,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"27","issue":"1","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":0.9994999766349792,"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":0.9994999766349792,"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":0.998199999332428,"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/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.9962000250816345,"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":0.7998709678649902},{"id":"https://openalex.org/keywords/lambda","display_name":"Lambda","score":0.7178816795349121},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.7082034945487976},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6843721866607666},{"id":"https://openalex.org/keywords/minor","display_name":"Minor (academic)","score":0.649101734161377},{"id":"https://openalex.org/keywords/degree","display_name":"Degree (music)","score":0.6489400863647461},{"id":"https://openalex.org/keywords/constant","display_name":"Constant (computer programming)","score":0.5994022488594055},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.5580461621284485},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.37659603357315063},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.20486152172088623},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.11419087648391724},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.0703936219215393},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.047924309968948364},{"id":"https://openalex.org/keywords/humanities","display_name":"Humanities","score":0.045909106731414795}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7998709678649902},{"id":"https://openalex.org/C2778113609","wikidata":"https://www.wikidata.org/wiki/Q10897","display_name":"Lambda","level":2,"score":0.7178816795349121},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.7082034945487976},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6843721866607666},{"id":"https://openalex.org/C2779760435","wikidata":"https://www.wikidata.org/wiki/Q5396169","display_name":"Minor (academic)","level":2,"score":0.649101734161377},{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.6489400863647461},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.5994022488594055},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.5580461621284485},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.37659603357315063},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.20486152172088623},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.11419087648391724},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.0703936219215393},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.047924309968948364},{"id":"https://openalex.org/C15708023","wikidata":"https://www.wikidata.org/wiki/Q80083","display_name":"Humanities","level":1,"score":0.045909106731414795},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","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/C24890656","wikidata":"https://www.wikidata.org/wiki/Q82811","display_name":"Acoustics","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.37236/8847","is_oa":true,"landing_page_url":"https://doi.org/10.37236/8847","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v27i2p4/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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1907.01202","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1907.01202","pdf_url":"https://arxiv.org/pdf/1907.01202","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"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"}],"best_oa_location":{"id":"doi:10.37236/8847","is_oa":true,"landing_page_url":"https://doi.org/10.37236/8847","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v27i2p4/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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G3551006755","display_name":null,"funder_award_id":"418520","funder_id":"https://openalex.org/F4320334593","funder_display_name":"Natural Sciences and Engineering Research Council of Canada"}],"funders":[{"id":"https://openalex.org/F4320334593","display_name":"Natural Sciences and Engineering Research Council of Canada","ror":"https://ror.org/01h531d29"},{"id":"https://openalex.org/F4320334704","display_name":"Australian Research Council","ror":"https://ror.org/05mmh0f86"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2956003067.pdf","grobid_xml":"https://content.openalex.org/works/W2956003067.grobid-xml"},"referenced_works_count":36,"referenced_works":["https://openalex.org/W48717235","https://openalex.org/W49540560","https://openalex.org/W1606591565","https://openalex.org/W1672577152","https://openalex.org/W1967546560","https://openalex.org/W1969934473","https://openalex.org/W1979717362","https://openalex.org/W1989340449","https://openalex.org/W1991445030","https://openalex.org/W1997241423","https://openalex.org/W2001021855","https://openalex.org/W2047993669","https://openalex.org/W2052409046","https://openalex.org/W2065163524","https://openalex.org/W2089714237","https://openalex.org/W2120650079","https://openalex.org/W2132418889","https://openalex.org/W2138147338","https://openalex.org/W2151438396","https://openalex.org/W2161040448","https://openalex.org/W2161786935","https://openalex.org/W2162070115","https://openalex.org/W2170136611","https://openalex.org/W2170758948","https://openalex.org/W2232156287","https://openalex.org/W2380988539","https://openalex.org/W2575334355","https://openalex.org/W2899702797","https://openalex.org/W2920971076","https://openalex.org/W2963017028","https://openalex.org/W2965696596","https://openalex.org/W3088736520","https://openalex.org/W3098569837","https://openalex.org/W3100674739","https://openalex.org/W4205775268","https://openalex.org/W4294557117"],"related_works":["https://openalex.org/W2368619281","https://openalex.org/W2377143628","https://openalex.org/W2355995962","https://openalex.org/W2392462513","https://openalex.org/W2362718971","https://openalex.org/W2368173763","https://openalex.org/W2977970519","https://openalex.org/W191560551","https://openalex.org/W2158905189","https://openalex.org/W4319874546"],"abstract_inverted_index":{"We":[0],"show":[1],"that":[2,24,72],"for":[3,8,53,60,76],"sufficiently":[4],"large":[5],"$d$":[6,35],"and":[7,32,59,66,81],"$t\\geq":[9],"d+1$,":[10],"there":[11],"is":[12,36,44,84],"a":[13,38,89],"graph":[14,27],"$G$":[15],"with":[16,29],"average":[17,33],"degree":[18,34],"$(1-\\varepsilon)\\lambda":[19],"t":[20],"\\sqrt{\\ln":[21],"d}$":[22],"such":[23],"almost":[25],"every":[26],"$H$":[28],"$t$":[30],"vertices":[31],"not":[37],"minor":[39],"of":[40],"$G$,":[41],"where":[42],"$\\lambda=0.63817\\dots$":[43],"an":[45,73],"explicitly":[46],"defined":[47],"constant.":[48],"This":[49],"generalises":[50],"analogous":[51],"results":[52],"complete":[54],"graphs":[55,63,78],"by":[56,64,79],"Thomason":[57,67],"(2001)":[58],"general":[61],"dense":[62],"Myers":[65],"(2005).":[68],"It":[69],"also":[70],"shows":[71],"upper":[74],"bound":[75],"sparse":[77],"Reed":[80],"Wood":[82],"(2016)":[83],"best":[85],"possible":[86],"up":[87],"to":[88],"constant":[90],"factor.":[91]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":2},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":1}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}