{"id":"https://openalex.org/W7220716599","doi":"https://doi.org/10.48550/arxiv.2610.04891","title":"A note on the injective edge coloring of graphs with bounded maximum degree","display_name":"A note on the injective edge coloring of graphs with bounded maximum degree","publication_year":2026,"publication_date":"2026-10-04","ids":{"openalex":"https://openalex.org/W7220716599","doi":"https://doi.org/10.48550/arxiv.2610.04891"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.04891","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.04891","pdf_url":"https://arxiv.org/pdf/2610.04891","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/2610.04891","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5046008005","display_name":"Lin Tian","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I169615421","display_name":"Middle Tennessee State University","ror":"https://ror.org/02n1hzn07","country_code":"US","type":"education","lineage":["https://openalex.org/I169615421"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Tian, Lin","raw_affiliation_strings":["Department of Mathematical Sciences , Middle Tennessee State University , Murfreesboro , TN 37132 , USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematical Sciences , Middle Tennessee State University , Murfreesboro , TN 37132 , USA","institution_ids":["https://openalex.org/I169615421"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5157747185","display_name":"Runze Wang","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I7956340","display_name":"Augustana College","ror":"https://ror.org/04bqrfs31","country_code":"US","type":"education","lineage":["https://openalex.org/I7956340"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Wang, Runze","raw_affiliation_strings":["Department of Mathematics and Computer Science , Augustana College , Rock Island , IL 61201 , USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Computer Science , Augustana College , Rock Island , IL 61201 , USA","institution_ids":["https://openalex.org/I7956340"]}]}],"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/T12541","display_name":"Graph Labeling and Dimension Problems","score":0.9990000128746033,"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":0.9990000128746033,"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/T10374","display_name":"Advanced Graph Theory Research","score":0.000699999975040555,"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":0.0,"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/maximum-degree","display_name":"maximum degree","score":0.7200000286102295},{"id":"https://openalex.org/keywords/graph-coloring","display_name":"graph coloring","score":0.6000000238418579},{"id":"https://openalex.org/keywords/upper-bounds","display_name":"upper bounds","score":0.5040000081062317}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.8043000102043152},{"id":"https://openalex.org/C128107574","wikidata":"https://www.wikidata.org/wiki/Q182003","display_name":"Injective function","level":2,"score":0.804099977016449},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8026000261306763},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.6241999864578247},{"id":"https://openalex.org/C123809776","wikidata":"https://www.wikidata.org/wiki/Q1050972","display_name":"Edge coloring","level":5,"score":0.6015999913215637},{"id":"https://openalex.org/C51658606","wikidata":"https://www.wikidata.org/wiki/Q512897","display_name":"Brooks' theorem","level":5,"score":0.5612000226974487},{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.54339998960495},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.5012000203132629},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.47290000319480896},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4390000104904175},{"id":"https://openalex.org/C196956537","wikidata":"https://www.wikidata.org/wiki/Q202021","display_name":"Chromatic scale","level":2,"score":0.43709999322891235},{"id":"https://openalex.org/C162307627","wikidata":"https://www.wikidata.org/wiki/Q204833","display_name":"Enhanced Data Rates for GSM Evolution","level":2,"score":0.41620001196861267},{"id":"https://openalex.org/C21642379","wikidata":"https://www.wikidata.org/wiki/Q3342988","display_name":"Complete coloring","level":5,"score":0.3294999897480011},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.28679999709129333},{"id":"https://openalex.org/C76946457","wikidata":"https://www.wikidata.org/wiki/Q504843","display_name":"Graph coloring","level":3,"score":0.2818000018596649},{"id":"https://openalex.org/C158319403","wikidata":"https://www.wikidata.org/wiki/Q5477527","display_name":"Fractional coloring","level":5,"score":0.2806999981403351}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.04891","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.04891","pdf_url":"https://arxiv.org/pdf/2610.04891","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.2610.04891","doi":"https://doi.org/10.48550/arxiv.2610.04891","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.04891","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:2610.04891","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.04891","pdf_url":"https://arxiv.org/pdf/2610.04891","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":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7220716599.pdf","grobid_xml":"https://content.openalex.org/works/W7220716599.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W7164006402","https://openalex.org/W4406272130","https://openalex.org/W4402408546","https://openalex.org/W3162029291","https://openalex.org/W4206161068","https://openalex.org/W7133354724","https://openalex.org/W4409458810","https://openalex.org/W4408443358","https://openalex.org/W4416571714","https://openalex.org/W7081580488"],"abstract_inverted_index":{"Let":[0,52],"$G$":[1,43],"be":[2,54],"a":[3,21],"graph.":[4],"Under":[5],"an":[6,45],"\\emph{injective":[7,28],"edge":[8,47],"coloring}":[9],"of":[10,31,58],"$G$,":[11,32],"any":[12],"two":[13],"edges":[14],"at":[15],"distance":[16],"$2$":[17],"or":[18],"belonging":[19],"to":[20],"common":[22],"triangle":[23],"receive":[24],"distinct":[25],"colors.":[26,51],"The":[27],"chromatic":[29],"index}":[30],"denoted":[33],"by":[34,74],"$\u03c7'_{inj}(G)$,":[35],"is":[36],"the":[37,55],"smallest":[38],"integer":[39],"$k$":[40,50],"such":[41],"that":[42,63,76],"admits":[44],"injective":[46],"coloring":[48],"with":[49],"$\u0394$":[53],"maximum":[56],"degree":[57],"$G$.":[59],"Ferdjallah":[60],"et":[61],"al.~proved":[62],"$\u03c7'_{inj}(G)":[64,77],"\\le":[65,78],"2(\u0394-1)^2$.":[66],"In":[67],"this":[68,72],"paper,":[69],"we":[70],"improve":[71],"bound":[73],"proving":[75],"2(\u0394-":[79],"1)^2":[80],"-\u0394+":[81],"3$":[82],"if":[83],"$\u0394\\ge":[84],"4$.":[85]},"counts_by_year":[],"updated_date":"2026-10-28T10:33:24.094632","created_date":"2026-10-07T00:00:00"}