{"id":"https://openalex.org/W7222610787","doi":"https://doi.org/10.48550/arxiv.2610.21506","title":"Strong coloring numbers of grids","display_name":"Strong coloring numbers of grids","publication_year":2026,"publication_date":"2026-10-15","ids":{"openalex":"https://openalex.org/W7222610787","doi":"https://doi.org/10.48550/arxiv.2610.21506"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.21506","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.21506","pdf_url":"https://arxiv.org/pdf/2610.21506","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2610.21506","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5107964690","display_name":"Piotr Micek","orcid":"https://orcid.org/0000-0001-9796-5375","observed_orcids":["https://orcid.org/0000-0001-9796-5375"]},"institutions":[{"id":"https://openalex.org/I126596746","display_name":"Jagiellonian University","ror":"https://ror.org/03bqmcz70","country_code":"PL","type":"education","lineage":["https://openalex.org/I126596746"]}],"countries":["PL"],"is_corresponding":false,"raw_author_name":"Micek, Piotr","raw_affiliation_strings":["(P. Micek) Theoretical Computer Science Department , Faculty of Mathematics and Computer Science , Jagiellonian University , Krak\u00f3w , Poland (","T. Ueckerdt)"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"(P. Micek) Theoretical Computer Science Department , Faculty of Mathematics and Computer Science , Jagiellonian University , Krak\u00f3w , Poland (","institution_ids":["https://openalex.org/I126596746"]},{"raw_affiliation_string":"T. Ueckerdt)","institution_ids":[]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5063296115","display_name":"Torsten Ueckerdt","orcid":"https://orcid.org/0000-0002-0645-9715","observed_orcids":["https://orcid.org/0000-0002-0645-9715"]},"institutions":[{"id":"https://openalex.org/I102335020","display_name":"Karlsruhe Institute of Technology","ror":"https://ror.org/04t3en479","country_code":"DE","type":"education","lineage":["https://openalex.org/I102335020","https://openalex.org/I1305996414"]},{"id":"https://openalex.org/I126596746","display_name":"Jagiellonian University","ror":"https://ror.org/03bqmcz70","country_code":"PL","type":"education","lineage":["https://openalex.org/I126596746"]}],"countries":["DE","PL"],"is_corresponding":false,"raw_author_name":"Ueckerdt, Torsten","raw_affiliation_strings":["(P. Micek) Theoretical Computer Science Department , Faculty of Mathematics and Computer Science , Jagiellonian University , Krak\u00f3w , Poland (","Institute of Theoretical Informatics , Karlsruhe Institute of Technology , Karl- sruhe , Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"(P. Micek) Theoretical Computer Science Department , Faculty of Mathematics and Computer Science , Jagiellonian University , Krak\u00f3w , Poland (","institution_ids":["https://openalex.org/I126596746"]},{"raw_affiliation_string":"Institute of Theoretical Informatics , Karlsruhe Institute of Technology , Karl- sruhe , Germany","institution_ids":["https://openalex.org/I102335020"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10374","display_name":"Advanced Graph Theory Research","score":0.7616999745368958,"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"}},"topics":[{"id":"https://openalex.org/T10374","display_name":"Advanced Graph Theory Research","score":0.7616999745368958,"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/T11329","display_name":"Limits and Structures in Graph Theory","score":0.22579999268054962,"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/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.002400000113993883,"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/grids","display_name":"grids","score":0.6570000052452087},{"id":"https://openalex.org/keywords/brambling","display_name":"Brambling","score":0.6029999852180481},{"id":"https://openalex.org/keywords/asymptotic-growth","display_name":"asymptotic growth","score":0.5740000009536743},{"id":"https://openalex.org/keywords/unit-ball","display_name":"unit ball","score":0.8339999914169312},{"id":"https://openalex.org/keywords/graph-coloring","display_name":"graph coloring","score":0.5600000023841858},{"id":"https://openalex.org/keywords/gpt-5-6","display_name":"GPT-5.6","score":0.6610000133514404}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.829800009727478},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.795199990272522},{"id":"https://openalex.org/C80899671","wikidata":"https://www.wikidata.org/wiki/Q1304193","display_name":"Vertex (graph theory)","level":3,"score":0.6263999938964844},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.5863000154495239},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.48410001397132874},{"id":"https://openalex.org/C187691185","wikidata":"https://www.wikidata.org/wiki/Q2020720","display_name":"Grid","level":2,"score":0.4749000072479248},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4034999907016754},{"id":"https://openalex.org/C2778307483","wikidata":"https://www.wikidata.org/wiki/Q5149038","display_name":"Colored","level":2,"score":0.3716999888420105},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.3549000024795532},{"id":"https://openalex.org/C2984853995","wikidata":"https://www.wikidata.org/wiki/Q1060343","display_name":"Independence number","level":3,"score":0.353300005197525},{"id":"https://openalex.org/C126875415","wikidata":"https://www.wikidata.org/wiki/Q203425","display_name":"Unit circle","level":2,"score":0.3452000021934509},{"id":"https://openalex.org/C178635117","wikidata":"https://www.wikidata.org/wiki/Q747499","display_name":"RADIUS","level":2,"score":0.329800009727478},{"id":"https://openalex.org/C158319403","wikidata":"https://www.wikidata.org/wiki/Q5477527","display_name":"Fractional coloring","level":5,"score":0.32749998569488525},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.31769999861717224},{"id":"https://openalex.org/C61797465","wikidata":"https://www.wikidata.org/wiki/Q1188986","display_name":"Term (time)","level":2,"score":0.2867000102996826},{"id":"https://openalex.org/C76946457","wikidata":"https://www.wikidata.org/wiki/Q504843","display_name":"Graph coloring","level":3,"score":0.28299999237060547},{"id":"https://openalex.org/C76444178","wikidata":"https://www.wikidata.org/wiki/Q72897900","display_name":"Connectivity","level":3,"score":0.2727000117301941},{"id":"https://openalex.org/C122637931","wikidata":"https://www.wikidata.org/wiki/Q118084","display_name":"Unit (ring theory)","level":2,"score":0.2700999975204468},{"id":"https://openalex.org/C200288055","wikidata":"https://www.wikidata.org/wiki/Q2621792","display_name":"Element (criminal law)","level":2,"score":0.25450000166893005}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.21506","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.21506","pdf_url":"https://arxiv.org/pdf/2610.21506","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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"},{"id":"doi:10.48550/arxiv.2610.21506","doi":"https://doi.org/10.48550/arxiv.2610.21506","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.21506","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2610.21506","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.21506","pdf_url":"https://arxiv.org/pdf/2610.21506","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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"},"sustainable_development_goals":[],"study_designs":[],"awards":[{"id":"https://openalex.org/G8652285958","display_name":"ProduktStruktur von Graphen","funder_award_id":"532213793","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"}],"funders":[{"id":"https://openalex.org/F4320320879","display_name":"Deutsche Forschungsgemeinschaft","ror":"https://ror.org/018mejw64"},{"id":"https://openalex.org/F4320322650","display_name":"Narodowym Centrum Nauki","ror":"https://ror.org/03ha2q922"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7222610787.pdf","grobid_xml":"https://content.openalex.org/works/W7222610787.grobid-xml"},"referenced_works_count":3,"referenced_works":["https://openalex.org/W2952466777","https://openalex.org/W3144518550","https://openalex.org/W4414295387"],"related_works":["https://openalex.org/W3144518550","https://openalex.org/W1976536763","https://openalex.org/W1583570361","https://openalex.org/W7098784097","https://openalex.org/W1666210965","https://openalex.org/W2808723058","https://openalex.org/W2759959783","https://openalex.org/W3017746715","https://openalex.org/W1661524359","https://openalex.org/W1943926878"],"abstract_inverted_index":{"For":[0,99],"positive":[1],"integers":[2],"$n$":[3,12,66],"and":[4,36],"$d$,":[5,48],"the":[6,14,40,49,55,100],"$d$-dimensional":[7],"grid":[8],"of":[9,54,60,93,109,115],"side":[10],"length":[11],"is":[13],"graph":[15],"$G(n,d)$":[16,106],"with":[17,112],"vertex":[18],"set":[19],"$[n]^d$,":[20],"in":[21,32,38,52,96,105],"which":[22,86],"two":[23],"vertices":[24],"are":[25],"adjacent":[26],"if":[27],"they":[28],"differ":[29],"by":[30,81,124],"$1$":[31],"exactly":[33],"one":[34],"coordinate":[35],"agree":[37],"all":[39],"other":[41],"coordinates.":[42],"We":[43],"determine,":[44],"for":[45,65,90],"every":[46],"fixed":[47],"asymptotic":[50],"growth":[51],"$r$":[53],"$r$th":[56],"strong":[57],"coloring":[58],"numbers":[59],"these":[61],"grids.":[62],"More":[63],"precisely":[64],"large":[67],"enough,":[68],"${\\rm":[69],"scol}_r(G(n,d))":[70],"=":[71],"\u03a9_d\\bigl(r^{d-1}\\bigr)$":[72],"as":[73],"$r\\to\\infty$.":[74],"This":[75,120],"matches":[76],"an":[77],"upper":[78],"bound":[79],"given":[80],"a":[82,107],"volume":[83],"packing":[84],"argument,":[85],"holds":[87],"more":[88],"generally":[89],"touching":[91],"graphs":[92],"unit":[94],"balls":[95],"$\\mathbb":[97],"R^d$.":[98],"lower":[101],"bound,":[102],"we":[103],"construct":[104],"bramble":[108],"order":[110],"$\u03a9_d(n^{d-1})$":[111],"each":[113],"element":[114],"radius":[116],"at":[117],"most":[118],"$dn$.":[119],"construction":[121],"was":[122],"found":[123],"OpenAI's":[125],"GPT-5.6":[126],"Sol.":[127]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-10-17T00:00:00"}