{"id":"https://openalex.org/W7214307644","doi":"https://doi.org/10.48550/arxiv.2609.29623","title":"On Alon's problem concerning the difference between connected domination number and domination number","display_name":"On Alon's problem concerning the difference between connected domination number and domination number","publication_year":2026,"publication_date":"2026-08-28","ids":{"openalex":"https://openalex.org/W7214307644","doi":"https://doi.org/10.48550/arxiv.2609.29623"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.29623","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.29623","pdf_url":"https://arxiv.org/pdf/2609.29623","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/2609.29623","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5152920566","display_name":"Chengli Li","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I66867065","display_name":"East China Normal University","ror":"https://ror.org/02n96ep67","country_code":"CN","type":"education","lineage":["https://openalex.org/I66867065"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Li, Chengli","raw_affiliation_strings":["School of Mathematical Sciences , East China Normal University , Shanghai 200241 , China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematical Sciences , East China Normal University , Shanghai 200241 , China","institution_ids":["https://openalex.org/I66867065"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5152884584","display_name":"Bo Zhou","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I187400657","display_name":"South China Normal University","ror":"https://ror.org/01kq0pv72","country_code":"CN","type":"education","lineage":["https://openalex.org/I187400657"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Zhou, Bo","raw_affiliation_strings":["School of Mathematical Sciences , South China Normal University , Guangzhou 510631 , China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematical Sciences , South China Normal University , Guangzhou 510631 , China","institution_ids":["https://openalex.org/I187400657"]}]}],"institutions":[],"countries_distinct_count":1,"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":0.8192999958992004,"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":0.8192999958992004,"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":0.16820000112056732,"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":0.0052999998442828655,"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/connected-domination-number","display_name":"connected domination number","score":0.8489999771118164},{"id":"https://openalex.org/keywords/domination-number","display_name":"domination number","score":0.9980000257492065},{"id":"https://openalex.org/keywords/minimum-degree","display_name":"minimum degree","score":0.5849999785423279},{"id":"https://openalex.org/keywords/asymptotic-estimates","display_name":"asymptotic bounds","score":0.8199999928474426},{"id":"https://openalex.org/keywords/extremal-graph-theory","display_name":"extremal graph theory","score":0.734000027179718},{"id":"https://openalex.org/keywords/connected-graphs","display_name":"connected graphs","score":0.5139999985694885},{"id":"https://openalex.org/keywords/domination-in-graphs","display_name":"graph domination","score":0.3869999945163727}],"concepts":[{"id":"https://openalex.org/C83833204","wikidata":"https://www.wikidata.org/wiki/Q5290322","display_name":"Domination analysis","level":4,"score":0.8781999945640564},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8321999907493591},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7554000020027161},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.5651000142097473},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.560699999332428},{"id":"https://openalex.org/C193435613","wikidata":"https://www.wikidata.org/wiki/Q2997928","display_name":"Connected component","level":2,"score":0.46700000762939453},{"id":"https://openalex.org/C76444178","wikidata":"https://www.wikidata.org/wiki/Q72897900","display_name":"Connectivity","level":3,"score":0.43860000371932983},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4375999867916107},{"id":"https://openalex.org/C146661039","wikidata":"https://www.wikidata.org/wiki/Q2915204","display_name":"Dominating set","level":4,"score":0.40459999442100525},{"id":"https://openalex.org/C38754835","wikidata":"https://www.wikidata.org/wiki/Q2003238","display_name":"Strongly connected component","level":2,"score":0.32899999618530273},{"id":"https://openalex.org/C88230418","wikidata":"https://www.wikidata.org/wiki/Q131476","display_name":"Graph theory","level":2,"score":0.3059999942779541},{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.29499998688697815},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.2535000145435333}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.29623","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.29623","pdf_url":"https://arxiv.org/pdf/2609.29623","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.2609.29623","doi":"https://doi.org/10.48550/arxiv.2609.29623","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.29623","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:2609.29623","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.29623","pdf_url":"https://arxiv.org/pdf/2609.29623","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/G6097193629","display_name":null,"funder_award_id":"12571364","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7214307644.pdf","grobid_xml":"https://content.openalex.org/works/W7214307644.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W71968967","https://openalex.org/W2050150370","https://openalex.org/W4416166874","https://openalex.org/W2963285256","https://openalex.org/W2039349395","https://openalex.org/W1553675387","https://openalex.org/W2019231595","https://openalex.org/W7101253939","https://openalex.org/W3213647466","https://openalex.org/W1999483848"],"abstract_inverted_index":{"For":[0,97],"a":[1,46],"connected":[2,14,27],"graph":[3],"$G$,":[4],"let":[5,99],"$\u03b3(G)$":[6],"and":[7,13,57,73,76,103],"$\u03b3_c(G)$":[8],"denote":[9],"its":[10],"domination":[11,15],"number":[12],"number,":[16],"respectively.":[17],"Let":[18],"$M(n,k)$":[19,52],"be":[20,63],"the":[21,67,71],"maximum":[22],"of":[23,30],"$\u03b3_c(G)-\u03b3(G)$":[24],"over":[25,114],"all":[26,115],"$n$-vertex":[28],"graphs":[29],"minimum":[31],"degree":[32],"at":[33],"least":[34],"$k$.":[35,93],"Alon":[36],"proved":[37],"that":[38,60],"\\[":[39,100,118],"2\\left\\lfloor\\frac{n}{k+1}\\right\\rfloor-O(1)\\le":[40],"M(n,k)<\\frac{n}{k+1}":[41],"\\bigl(\\log\\lceil\\log(k+1)\\rceil+3\\bigr).":[42],"\\]":[43,110,122],"He":[44],"proposed":[45],"problem":[47,89],"to":[48,65,87],"determine":[49],"or":[50,79],"estimate":[51],"for":[53,90],"$n-1\\ge":[54],"k\\ge":[55],"3$,":[56],"particularly":[58],"remarked":[59],"it":[61],"would":[62],"interesting":[64],"close":[66],"$\\log\\log(k+1)$":[68],"gap":[69],"between":[70],"upper":[72],"lower":[74],"bounds":[75],"decide":[77],"whether":[78],"not":[80],"$M(n,k)=\u0398\\bigl(\\frac{n}{k+1}\\bigr)$.":[81],"We":[82],"give":[83],"an":[84],"asymptotic":[85],"answer":[86],"Alon's":[88],"sufficiently":[91],"large":[92],"More":[94],"precisely,":[95],"$M(k+1,k)=0$.":[96],"$n>k+1$,":[98,117],"\u03bd=\\frac{n}{k+1}":[101],"\\text{":[102],"}":[104],"\u03a6(x)=":[105],"\\frac{1}{x\\log\\!\\frac{x}{x-1}}":[106],"\\":[107],"(\\text{for":[108],"}x>1).":[109],"As":[111],"$k\\to\\infty$,":[112],"uniformly":[113],"integers":[116],"M(n,k)=":[119],"\\big(\u03a6(\u03bd)+o(1)\\big)":[120],"\u03bd\\log\\log(k+1).":[121]},"counts_by_year":[],"updated_date":"2026-11-06T12:05:46.655746","created_date":"2026-09-26T00:00:00"}
