{"id":"https://openalex.org/W7214731599","doi":"https://doi.org/10.48550/arxiv.2609.32903","title":"On the Minimum Possible Maximum Degree of Induced Subgraphs of Product Graphs","display_name":"On the Minimum Possible Maximum Degree of Induced Subgraphs of Product Graphs","publication_year":2026,"publication_date":"2026-09-26","ids":{"openalex":"https://openalex.org/W7214731599","doi":"https://doi.org/10.48550/arxiv.2609.32903"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.32903","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.32903","pdf_url":"https://arxiv.org/pdf/2609.32903","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/2609.32903","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5123706413","display_name":"Elizaveta Popova","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I53964585","display_name":"Weizmann Institute of Science","ror":"https://ror.org/0316ej306","country_code":"IL","type":"education","lineage":["https://openalex.org/I53964585"]}],"countries":["IL"],"is_corresponding":true,"raw_author_name":"Popova, Elizaveta","raw_affiliation_strings":["Weizmann Institute of Science , Israel"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Weizmann Institute of Science , Israel","institution_ids":["https://openalex.org/I53964585"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5123706413"],"corresponding_institution_ids":["https://openalex.org/I53964585"],"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/T11329","display_name":"Limits and Structures in Graph Theory","score":0.9965999722480774,"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":0.9965999722480774,"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":0.0026000000070780516,"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/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.0003000000142492354,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/induced-subgraphs","display_name":"induced subgraphs","score":0.9330000281333923},{"id":"https://openalex.org/keywords/maximum-degree","display_name":"maximum degree","score":0.8640000224113464},{"id":"https://openalex.org/keywords/tensor-product-of-graphs","display_name":"tensor product of graphs","score":0.8159999847412109},{"id":"https://openalex.org/keywords/hypercube","display_name":"hypercube","score":0.9179999828338623},{"id":"https://openalex.org/keywords/average-degree","display_name":"average degree","score":0.6259999871253967},{"id":"https://openalex.org/keywords/minimum-degree","display_name":"minimum degree","score":0.49799999594688416},{"id":"https://openalex.org/keywords/independent-sets","display_name":"independent sets","score":0.5799999833106995},{"id":"https://openalex.org/keywords/complete-graphs","display_name":"complete graphs","score":0.5519999861717224}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8252999782562256},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.8245999813079834},{"id":"https://openalex.org/C50820777","wikidata":"https://www.wikidata.org/wiki/Q213723","display_name":"Hypercube","level":2,"score":0.7408000230789185},{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.6601999998092651},{"id":"https://openalex.org/C51255310","wikidata":"https://www.wikidata.org/wiki/Q1163016","display_name":"Tensor product","level":2,"score":0.5629000067710876},{"id":"https://openalex.org/C29696283","wikidata":"https://www.wikidata.org/wiki/Q369440","display_name":"Hamming graph","level":5,"score":0.521399974822998},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.49799999594688416},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.47769999504089355},{"id":"https://openalex.org/C193319292","wikidata":"https://www.wikidata.org/wiki/Q272172","display_name":"Hamming distance","level":2,"score":0.46630001068115234},{"id":"https://openalex.org/C90673727","wikidata":"https://www.wikidata.org/wiki/Q901718","display_name":"Product (mathematics)","level":2,"score":0.4189000129699707},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.3718999922275543},{"id":"https://openalex.org/C73150493","wikidata":"https://www.wikidata.org/wiki/Q853922","display_name":"Hamming code","level":4,"score":0.3675000071525574},{"id":"https://openalex.org/C2778012994","wikidata":"https://www.wikidata.org/wiki/Q24743362","display_name":"Induced subgraph","level":4,"score":0.351500004529953},{"id":"https://openalex.org/C2993105083","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Simple graph","level":3,"score":0.32820001244544983},{"id":"https://openalex.org/C76444178","wikidata":"https://www.wikidata.org/wiki/Q72897900","display_name":"Connectivity","level":3,"score":0.3163999915122986},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.3122999966144562},{"id":"https://openalex.org/C58228413","wikidata":"https://www.wikidata.org/wiki/Q15704639","display_name":"Induced path","level":5,"score":0.30090001225471497},{"id":"https://openalex.org/C36038622","wikidata":"https://www.wikidata.org/wiki/Q5428703","display_name":"Factor-critical graph","level":5,"score":0.27399998903274536},{"id":"https://openalex.org/C88230418","wikidata":"https://www.wikidata.org/wiki/Q131476","display_name":"Graph theory","level":2,"score":0.25110000371932983}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.32903","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.32903","pdf_url":"https://arxiv.org/pdf/2609.32903","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.2609.32903","doi":"https://doi.org/10.48550/arxiv.2609.32903","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.32903","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.32903","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.32903","pdf_url":"https://arxiv.org/pdf/2609.32903","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/G7409033375","display_name":null,"funder_award_id":"2364/25","funder_id":"https://openalex.org/F4320322252","funder_display_name":"Israel Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320322252","display_name":"Israel Science Foundation","ror":"https://ror.org/04sazxf24"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7214731599.pdf","grobid_xml":"https://content.openalex.org/works/W7214731599.grobid-xml"},"referenced_works_count":7,"referenced_works":["https://openalex.org/W2125710943","https://openalex.org/W3012453944","https://openalex.org/W3013011289","https://openalex.org/W3043210608","https://openalex.org/W3186821417","https://openalex.org/W4399061718","https://openalex.org/W4408444842"],"related_works":["https://openalex.org/W2992307599","https://openalex.org/W3170780379","https://openalex.org/W4396809057","https://openalex.org/W6940057648","https://openalex.org/W3186821417","https://openalex.org/W4399061718","https://openalex.org/W2166264168","https://openalex.org/W4300621634","https://openalex.org/W7220948195","https://openalex.org/W2901484702"],"abstract_inverted_index":{"The":[0,43],"following":[1],"is":[2,46,92,103,166],"a":[3,9,24,39,115,133],"natural":[4],"and":[5,86,158],"fundamental":[6],"question":[7,100],"for":[8,154],"graph":[10,189],"$G$:":[11],"if":[12],"an":[13,129,186],"induced":[14,66,130],"subgraph":[15,67,131],"$H$":[16,37],"of":[17,36,41,48,57,68,114,128,132,179],"$G$":[18,102],"has":[19,77],"$x$":[20],"more":[21,73],"vertices":[22,76],"than":[23,74],"maximum":[25,34,78,164],"independent":[26],"set,":[27],"what":[28],"can":[29],"be":[30],"said":[31],"about":[32],"the":[33,58,69,105,110,122,140,148,161,170,180,183],"degree":[35,79,127,165],"as":[38],"function":[40],"$x$?":[42],"case":[44],"$x=1$":[45],"already":[47],"considerable":[49],"interest.":[50],"For":[51,117,147],"example,":[52],"in":[53],"his":[54],"celebrated":[55],"proof":[56],"sensitivity":[59],"conjecture,":[60],"Hao":[61],"Huang":[62],"showed":[63],"that":[64,89,139],"every":[65],"hypercube":[70],"$Q_n$":[71],"on":[72],"$2^{n-1}$":[75],"at":[80],"least":[81],"$\\sqrt{n}$.":[82],"Chung,":[83],"F\u00faredi,":[84],"Graham,":[85],"Seymour":[87],"proved":[88],"this":[90,95,99],"bound":[91],"tight.":[93],"In":[94],"paper,":[96],"we":[97,120,151,176],"study":[98],"when":[101],"either":[104],"$n$-fold":[106,111],"Hamming":[107,149],"product":[108,113,142],"or":[109],"tensor":[112,141],"triangle.":[116],"both":[118],"graphs,":[119],"determine":[121],"exact":[123],"minimum":[124,162,171],"possible":[125,163,172],"average":[126,173],"prescribed":[134],"size.":[135],"We":[136],"also":[137],"prove":[138],"exhibits":[143],"several":[144,155,178],"Huang-like":[145],"phenomena.":[146],"product,":[150],"show":[152],"that,":[153],"size":[156],"densities":[157],"large":[159],"$n$,":[160],"asymptotically":[167],"equal":[168],"to":[169,185],"degree.":[174],"Finally,":[175],"extend":[177],"results":[181],"from":[182],"triangle":[184],"arbitrary":[187],"complete":[188],"$K_k$.":[190]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-30T00:00:00"}