{"id":"https://openalex.org/W7214716139","doi":"https://doi.org/10.48550/arxiv.2609.32161","title":"Weak $k$-metric dimension of Hamming graphs: rectangular products and near-maximum parameters","display_name":"Weak $k$-metric dimension of Hamming graphs: rectangular products and near-maximum parameters","publication_year":2026,"publication_date":"2026-09-26","ids":{"openalex":"https://openalex.org/W7214716139","doi":"https://doi.org/10.48550/arxiv.2609.32161"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.32161","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.32161","pdf_url":"https://arxiv.org/pdf/2609.32161","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/2609.32161","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5138526478","display_name":"Aryan Kumar","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I51504820","display_name":"San Jose State University","ror":"https://ror.org/04qyvz380","country_code":"US","type":"education","lineage":["https://openalex.org/I51504820"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Kumar, Aryan","raw_affiliation_strings":["Department of Mathematics and Statistics San Jos\u00e9 State University San Jos\u00e9 , California , USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Statistics San Jos\u00e9 State University San Jos\u00e9 , California , USA","institution_ids":["https://openalex.org/I51504820"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5138526478"],"corresponding_institution_ids":["https://openalex.org/I51504820"],"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/T12541","display_name":"Graph Labeling and Dimension Problems","score":0.9564999938011169,"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/T12541","display_name":"Graph Labeling and Dimension Problems","score":0.9564999938011169,"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/T11476","display_name":"Graph theory and applications","score":0.012900000438094139,"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":0.00839999970048666,"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/hamming-graphs","display_name":"Hamming graphs","score":0.9679999947547913},{"id":"https://openalex.org/keywords/complete-bipartite-graphs","display_name":"complete bipartite graphs","score":0.8659999966621399},{"id":"https://openalex.org/keywords/hypercube","display_name":"hypercube","score":0.9909999966621399},{"id":"https://openalex.org/keywords/metric-dimension","display_name":"metric dimension","score":0.41200000047683716}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8266000151634216},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.7067000269889832},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.6797999739646912},{"id":"https://openalex.org/C193319292","wikidata":"https://www.wikidata.org/wiki/Q272172","display_name":"Hamming distance","level":2,"score":0.6751999855041504},{"id":"https://openalex.org/C50820777","wikidata":"https://www.wikidata.org/wiki/Q213723","display_name":"Hypercube","level":2,"score":0.6610000133514404},{"id":"https://openalex.org/C73150493","wikidata":"https://www.wikidata.org/wiki/Q853922","display_name":"Hamming code","level":4,"score":0.6536999940872192},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5755000114440918},{"id":"https://openalex.org/C111335779","wikidata":"https://www.wikidata.org/wiki/Q3454686","display_name":"Reduction (mathematics)","level":2,"score":0.489300012588501},{"id":"https://openalex.org/C29696283","wikidata":"https://www.wikidata.org/wiki/Q369440","display_name":"Hamming graph","level":5,"score":0.4763999879360199},{"id":"https://openalex.org/C63361517","wikidata":"https://www.wikidata.org/wiki/Q5645805","display_name":"Hamming weight","level":5,"score":0.4537000060081482},{"id":"https://openalex.org/C48372109","wikidata":"https://www.wikidata.org/wiki/Q3913","display_name":"Binary number","level":2,"score":0.4505000114440918},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4219000041484833},{"id":"https://openalex.org/C166530166","wikidata":"https://www.wikidata.org/wiki/Q1970635","display_name":"Hamming bound","level":5,"score":0.38350000977516174},{"id":"https://openalex.org/C112932734","wikidata":"https://www.wikidata.org/wiki/Q2322550","display_name":"Hamming(7,4)","level":5,"score":0.3208000063896179},{"id":"https://openalex.org/C63435697","wikidata":"https://www.wikidata.org/wiki/Q864135","display_name":"Binary code","level":3,"score":0.3140999972820282},{"id":"https://openalex.org/C187455244","wikidata":"https://www.wikidata.org/wiki/Q942353","display_name":"Boolean function","level":2,"score":0.2818000018596649},{"id":"https://openalex.org/C70518039","wikidata":"https://www.wikidata.org/wiki/Q16000077","display_name":"Dimensionality reduction","level":2,"score":0.2718999981880188},{"id":"https://openalex.org/C165064840","wikidata":"https://www.wikidata.org/wiki/Q1321061","display_name":"Matching (statistics)","level":2,"score":0.2563000023365021}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.32161","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.32161","pdf_url":"https://arxiv.org/pdf/2609.32161","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.2609.32161","doi":"https://doi.org/10.48550/arxiv.2609.32161","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.32161","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:2609.32161","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.32161","pdf_url":"https://arxiv.org/pdf/2609.32161","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":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7214716139.pdf","grobid_xml":"https://content.openalex.org/works/W7214716139.grobid-xml"},"referenced_works_count":1,"referenced_works":["https://openalex.org/W7162131307"],"related_works":["https://openalex.org/W7135098360","https://openalex.org/W7162131307","https://openalex.org/W2040179039","https://openalex.org/W4394924821","https://openalex.org/W4386552592","https://openalex.org/W4289785297","https://openalex.org/W7151754528","https://openalex.org/W2795067783","https://openalex.org/W7124927250","https://openalex.org/W2108442028"],"abstract_inverted_index":{"We":[0],"determine":[1,111],"weak":[2,62],"$k$-metric":[3],"dimensions":[4],"for":[5,77,93],"several":[6],"families":[7],"of":[8,23],"Hamming":[9],"graphs.":[10],"For":[11,34,47,83,102],"$n\\ge3$,":[12],"$m>n$,":[13],"and":[14,28,58,85,95,106],"$3\\le":[15],"k\\le2n$,":[16],"a":[17,69],"cyclic":[18],"construction":[19],"proves":[20],"Conjecture":[21],"6.1":[22],"Fern\u00e1ndez,":[24],"Klav\u017ear,":[25],"Kuziak,":[26],"Mu\u00f1oz-M\u00e1rquez":[27],"Yero":[29],"(2026)":[30],"on":[31],"$K_n\\square":[32],"K_m$.":[33],"these":[35],"rectangular":[36],"products,":[37],"we":[38,52,87,109],"also":[39,110],"prove":[40,88],"that":[41,54,97],"$\\operatorname{wdim}_2(K_n\\square":[42],"K_m)=m$":[43],"exactly":[44],"when":[45],"$m\\ge2n-2$.":[46],"hypercubes":[48],"$Q_d$":[49],"with":[50,119],"$d\\ge2$,":[51],"show":[53,96],"consecutive":[55],"parameters":[56],"$2s-1$":[57],"$2s$":[59],"have":[60],"identical":[61],"resolving":[63],"sets.":[64],"Near":[65],"the":[66,89],"maximum":[67],"parameter,":[68],"reduction":[70],"to":[71],"restricted-distance":[72],"binary":[73],"codes":[74],"determines":[75],"$\\operatorname{wdim}_{2^d-t}(Q_d)$":[76],"every":[78],"feasible":[79],"deficit":[80,107],"$0\\le":[81],"t\\le15$.":[82],"$L\\ge1$":[84],"$t\\in\\{2L,2L+1\\}$,":[86],"stabilization":[90],"formula":[91],"$\\operatorname{wdim}_{2^d-t}(Q_d)=2^d-L$":[92],"$d\\ge2^L+1$,":[94],"this":[98],"threshold":[99],"is":[100],"sharp.":[101],"each":[103],"fixed":[104],"$q\\ge3$":[105],"$t$,":[108],"$\\operatorname{wdim}_{2q^{d-1}-t}(K_q^{\\square":[112],"d})$":[113],"in":[114],"all":[115],"sufficiently":[116],"large":[117],"dimensions,":[118],"an":[120],"explicit":[121],"sufficient":[122],"condition.":[123]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-30T00:00:00"}