{"id":"https://openalex.org/W1719252778","doi":"https://doi.org/10.37236/640","title":"The $t$-Pebbling Number is Eventually Linear in $t$","display_name":"The $t$-Pebbling Number is Eventually Linear in $t$","publication_year":2011,"publication_date":"2011-07-22","ids":{"openalex":"https://openalex.org/W1719252778","doi":"https://doi.org/10.37236/640","mag":"1719252778"},"language":"en","primary_location":{"id":"doi:10.37236/640","is_oa":true,"landing_page_url":"https://doi.org/10.37236/640","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v18i1p153/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/v18i1p153/pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5103088991","display_name":"Michael M. Hoffmann","orcid":"https://orcid.org/0000-0001-5307-7106"},"institutions":[{"id":"https://openalex.org/I35440088","display_name":"ETH Zurich","ror":"https://ror.org/05a28rw58","country_code":"CH","type":"education","lineage":["https://openalex.org/I2799323385","https://openalex.org/I35440088"]}],"countries":["CH"],"is_corresponding":true,"raw_author_name":"Michael Hoffmann","raw_affiliation_strings":["[ETH, Zurich]"],"affiliations":[{"raw_affiliation_string":"[ETH, Zurich]","institution_ids":["https://openalex.org/I35440088"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5070164050","display_name":"Ji\u0159\u0131\u0301 Matou\u0161ek","orcid":"https://orcid.org/0000-0002-9447-0366"},"institutions":[{"id":"https://openalex.org/I21250087","display_name":"Charles University","ror":"https://ror.org/024d6js02","country_code":"CZ","type":"education","lineage":["https://openalex.org/I21250087"]}],"countries":["CZ"],"is_corresponding":false,"raw_author_name":"Ji\u0159\u00ed Matou\u0161ek","raw_affiliation_strings":["Charles University in Prague"],"affiliations":[{"raw_affiliation_string":"Charles University in Prague","institution_ids":["https://openalex.org/I21250087"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5080521209","display_name":"Yoshio Okamoto","orcid":"https://orcid.org/0000-0002-9826-7074"},"institutions":[{"id":"https://openalex.org/I177738480","display_name":"Japan Advanced Institute of Science and Technology","ror":"https://ror.org/03frj4r98","country_code":"JP","type":"education","lineage":["https://openalex.org/I177738480"]}],"countries":["JP"],"is_corresponding":false,"raw_author_name":"Yoshio Okamoto","raw_affiliation_strings":["Japan Advanced Inst. of Science and Technology"],"affiliations":[{"raw_affiliation_string":"Japan Advanced Inst. of Science and Technology","institution_ids":["https://openalex.org/I177738480"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5030467731","display_name":"Philipp Zumstein","orcid":"https://orcid.org/0000-0002-6485-9434"},"institutions":[{"id":"https://openalex.org/I35440088","display_name":"ETH Zurich","ror":"https://ror.org/05a28rw58","country_code":"CH","type":"education","lineage":["https://openalex.org/I2799323385","https://openalex.org/I35440088"]}],"countries":["CH"],"is_corresponding":false,"raw_author_name":"Philipp Zumstein","raw_affiliation_strings":["[ETH, Zurich]"],"affiliations":[{"raw_affiliation_string":"[ETH, Zurich]","institution_ids":["https://openalex.org/I35440088"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5103088991"],"corresponding_institution_ids":["https://openalex.org/I35440088"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":true,"cited_by_count":1,"citation_normalized_percentile":{"value":0.06390274,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"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/T11329","display_name":"Limits and Structures in Graph Theory","score":0.9980999827384949,"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":0.9980999827384949,"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/T11151","display_name":"Advanced Topology and Set Theory","score":0.9843999743461609,"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":0.9836000204086304,"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":0.899158239364624},{"id":"https://openalex.org/keywords/vertex","display_name":"Vertex (graph theory)","score":0.7583973407745361},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7551218867301941},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.6781593561172485},{"id":"https://openalex.org/keywords/omega","display_name":"Omega","score":0.5065332651138306},{"id":"https://openalex.org/keywords/integer","display_name":"Integer (computer science)","score":0.4395739436149597},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.404651939868927},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.14627328515052795},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.07387912273406982}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.899158239364624},{"id":"https://openalex.org/C80899671","wikidata":"https://www.wikidata.org/wiki/Q1304193","display_name":"Vertex (graph theory)","level":3,"score":0.7583973407745361},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7551218867301941},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.6781593561172485},{"id":"https://openalex.org/C2779557605","wikidata":"https://www.wikidata.org/wiki/Q9890","display_name":"Omega","level":2,"score":0.5065332651138306},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.4395739436149597},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.404651939868927},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.14627328515052795},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.07387912273406982},{"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/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.37236/640","is_oa":true,"landing_page_url":"https://doi.org/10.37236/640","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v18i1p153/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:10.1.1.372.5692","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.372.5692","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/v18i1p153/pdf/","raw_type":"text"}],"best_oa_location":{"id":"doi:10.37236/640","is_oa":true,"landing_page_url":"https://doi.org/10.37236/640","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v18i1p153/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":[{"id":"https://openalex.org/G2983449479","display_name":null,"funder_award_id":"Scientific Research","funder_id":"https://openalex.org/F4320334764","funder_display_name":"Japan Society for the Promotion of Science"},{"id":"https://openalex.org/G3459562248","display_name":null,"funder_award_id":"Grant","funder_id":"https://openalex.org/F4320334764","funder_display_name":"Japan Society for the Promotion of Science"},{"id":"https://openalex.org/G7752643416","display_name":null,"funder_award_id":"Japan","funder_id":"https://openalex.org/F4320334764","funder_display_name":"Japan Society for the Promotion of Science"}],"funders":[{"id":"https://openalex.org/F4320334764","display_name":"Japan Society for the Promotion of Science","ror":"https://ror.org/00hhkn466"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W1719252778.pdf","grobid_xml":"https://content.openalex.org/works/W1719252778.grobid-xml"},"referenced_works_count":8,"referenced_works":["https://openalex.org/W1523681510","https://openalex.org/W1899366296","https://openalex.org/W1979683483","https://openalex.org/W2013252038","https://openalex.org/W2122185950","https://openalex.org/W2136116874","https://openalex.org/W4255891826","https://openalex.org/W6645206086"],"related_works":["https://openalex.org/W238609530","https://openalex.org/W2734192830","https://openalex.org/W3016395638","https://openalex.org/W1636163329","https://openalex.org/W4292947165","https://openalex.org/W3045121788","https://openalex.org/W3201655533","https://openalex.org/W3134442314","https://openalex.org/W2281248943","https://openalex.org/W3102538728"],"abstract_inverted_index":{"In":[0],"graph":[1,41,91],"pebbling":[2,18,68,132],"games,":[3],"one":[4,26,30,145],"considers":[5],"a":[6,14,17,40,65,72,82,131],"distribution":[7,52],"of":[8,13,21,39,53,67,84],"pebbles":[9,24,55,78,140],"on":[10,31,56],"the":[11,44],"vertices":[12],"graph,":[15],"and":[16,28,58,130,143],"move":[19,133],"consists":[20],"taking":[22],"two":[23],"off":[25],"vertex":[27,61],"placing":[29],"an":[32],"adjacent":[33],"vertex.":[34],"The":[35],"$t$-pebbling":[36],"number":[37],"$\\pi_t(G)$":[38,93],"$G$":[42],"is":[43,94,114],"smallest":[45],"$m$":[46,54],"such":[47,105],"that":[48,88,99,106],"for":[49,89,108,117],"every":[50,59,90,121],"initial":[51],"$V(G)$":[57],"target":[60],"$x$":[62],"there":[63,101],"exists":[64],"sequence":[66],"moves":[69],"leading":[70],"to":[71,136,147],"distibution":[73],"with":[74],"at":[75,79,141],"least":[76],"$t$":[77],"$x$.":[80],"Answering":[81],"question":[83],"Sieben,":[85],"we":[86],"show":[87],"$G$,":[92],"eventually":[95],"linear":[96],"in":[97],"$t$;":[98],"is,":[100],"are":[102],"numbers":[103],"$a,b,t_0$":[104],"$\\pi_t(G)=at+b$":[107],"all":[109],"$t\\ge":[110],"t_0$.":[111],"Our":[112],"result":[113],"also":[115],"valid":[116],"weighted":[118],"graphs,":[119],"where":[120],"edge":[122],"$e=\\{u,v\\}$":[123],"has":[124],"some":[125],"integer":[126],"weight":[127],"$\\omega(e)\\ge":[128],"2$,":[129],"from":[134],"$u$":[135,142],"$v$":[137],"removes":[138],"$\\omega(e)$":[139],"adds":[144],"pebble":[146],"$v$.":[148]},"counts_by_year":[{"year":2024,"cited_by_count":1}],"updated_date":"2026-04-10T15:06:20.359241","created_date":"2025-10-10T00:00:00"}