{"id":"https://openalex.org/W7225558165","doi":"https://doi.org/10.20944/preprints202611.0784.v1","title":"Six Colors for Generalized Sierpi\u0144ski Graphs Over C5 and a Sharp Classification of Path Products","display_name":"Six Colors for Generalized Sierpi\u0144ski Graphs Over C5 and a Sharp Classification of Path Products","publication_year":2026,"publication_date":"2026-11-09","ids":{"openalex":"https://openalex.org/W7225558165","doi":"https://doi.org/10.20944/preprints202611.0784.v1"},"language":"en","primary_location":{"id":"doi:10.20944/preprints202611.0784.v1","doi":"https://doi.org/10.20944/preprints202611.0784.v1","is_oa":true,"landing_page_url":"https://doi.org/10.20944/preprints202611.0784.v1","pdf_url":null,"source":{"id":"https://openalex.org/S6309402219","display_name":"Preprints.org","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":null,"host_organization_name":null,"host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"posted-content"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.20944/preprints202611.0784.v1","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5160208694","display_name":"Xinyao Xu","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Xinyao Xu","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5160208694"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"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":0.600600004196167,"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":0.600600004196167,"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/T12541","display_name":"Graph Labeling and Dimension Problems","score":0.3621000051498413,"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":0.0044999998062849045,"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/packing-chromatic-number","display_name":"packing chromatic number","score":0.9879999756813049},{"id":"https://openalex.org/keywords/graph-coloring","display_name":"graph coloring","score":0.47600001096725464},{"id":"https://openalex.org/keywords/graph-templates","display_name":"graph templates","score":0.328000009059906}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8320000171661377},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7576000094413757},{"id":"https://openalex.org/C196956537","wikidata":"https://www.wikidata.org/wiki/Q202021","display_name":"Chromatic scale","level":2,"score":0.7081000208854675},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.6274999976158142},{"id":"https://openalex.org/C39927690","wikidata":"https://www.wikidata.org/wiki/Q11197","display_name":"Logarithm","level":2,"score":0.5098000168800354},{"id":"https://openalex.org/C2777735758","wikidata":"https://www.wikidata.org/wiki/Q817765","display_name":"Path (computing)","level":2,"score":0.49219998717308044},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.4805999994277954},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.42250001430511475},{"id":"https://openalex.org/C146834321","wikidata":"https://www.wikidata.org/wiki/Q2979672","display_name":"Closure (psychology)","level":2,"score":0.41940000653266907},{"id":"https://openalex.org/C45374587","wikidata":"https://www.wikidata.org/wiki/Q12525525","display_name":"Computation","level":2,"score":0.4043999910354614},{"id":"https://openalex.org/C204323151","wikidata":"https://www.wikidata.org/wiki/Q905424","display_name":"Range (aeronautics)","level":2,"score":0.3555999994277954},{"id":"https://openalex.org/C151201525","wikidata":"https://www.wikidata.org/wiki/Q177239","display_name":"Limit (mathematics)","level":2,"score":0.3443000018596649},{"id":"https://openalex.org/C162392398","wikidata":"https://www.wikidata.org/wiki/Q272404","display_name":"Finite set","level":2,"score":0.3319000005722046},{"id":"https://openalex.org/C48103436","wikidata":"https://www.wikidata.org/wiki/Q599031","display_name":"State (computer science)","level":2,"score":0.3174000084400177},{"id":"https://openalex.org/C119322782","wikidata":"https://www.wikidata.org/wiki/Q2662236","display_name":"VC dimension","level":2,"score":0.2728999853134155},{"id":"https://openalex.org/C64543145","wikidata":"https://www.wikidata.org/wiki/Q162942","display_name":"Intersection (aeronautics)","level":2,"score":0.26030001044273376},{"id":"https://openalex.org/C17231256","wikidata":"https://www.wikidata.org/wiki/Q5156540","display_name":"Completeness (order theory)","level":2,"score":0.2549000084400177},{"id":"https://openalex.org/C80661125","wikidata":"https://www.wikidata.org/wiki/Q5404299","display_name":"Packing dimension","level":5,"score":0.25429999828338623}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.20944/preprints202611.0784.v1","doi":"https://doi.org/10.20944/preprints202611.0784.v1","is_oa":true,"landing_page_url":"https://doi.org/10.20944/preprints202611.0784.v1","pdf_url":null,"source":{"id":"https://openalex.org/S6309402219","display_name":"Preprints.org","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":null,"host_organization_name":null,"host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"posted-content"}],"best_oa_location":{"id":"doi:10.20944/preprints202611.0784.v1","doi":"https://doi.org/10.20944/preprints202611.0784.v1","is_oa":true,"landing_page_url":"https://doi.org/10.20944/preprints202611.0784.v1","pdf_url":null,"source":{"id":"https://openalex.org/S6309402219","display_name":"Preprints.org","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":null,"host_organization_name":null,"host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"posted-content"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W2894111415","https://openalex.org/W4416103352","https://openalex.org/W2767574922","https://openalex.org/W2247215154","https://openalex.org/W1988724208","https://openalex.org/W4399198590","https://openalex.org/W4416359481","https://openalex.org/W3082064622","https://openalex.org/W1971582944","https://openalex.org/W4415239843"],"abstract_inverted_index":{"We":[0],"prove":[1],"that":[2],"the":[3,8,13,25,29,52,93,99,173,178],"packing":[4,95],"chromatic":[5,96],"number":[6,97],"of":[7,33,83,151],"generalized":[9],"Sierpi\u0144ski":[10,88],"graph":[11],"over":[12],"five-cycle":[14],"is":[15,110],"six":[16,121,158],"in":[17,70],"every":[18],"dimension":[19,71],"at":[20,35,62,84,133,138,153],"least":[21,36,63,85,154],"two.":[22],"Together":[23],"with":[24,132],"known":[26],"values":[27],"for":[28,130,157],"other":[30],"cycle":[31],"bases":[32],"length":[34,152],"four,":[37],"this":[38],"completes":[39],"their":[40,75],"classification.":[41],"Two":[42],"explicit":[43],"125-vertex":[44],"templates":[45],"and":[46,120,183],"twenty":[47],"oriented":[48],"profile":[49],"states":[50],"provide":[51],"construction.":[53],"Each":[54],"state":[55],"admits":[56],"two":[57,126,135,149],"independently":[58],"selectable":[59],"productions,":[60],"yielding":[61],"20":[64],"\u00b7":[65],"2^((5^(n\u22123)\u22121)/4)":[66],"optimal":[67],"labelled":[68],"colorings":[69],"n":[72,106,117],"\u2265":[73,104,107,115,118],"3;":[74],"normalized":[76],"base-two":[77],"logarithmic":[78],"count":[79],"has":[80],"a":[81],"limit":[82],"1/500.":[86],"For":[87],"path":[89],"products,":[90],"we":[91],"determine":[92],"maximum":[94],"throughout":[98],"previously":[100],"unresolved":[101],"range":[102],"m":[103,114],"12,":[105],"3:":[108],"it":[109],"seven":[111],"exactly":[112],"when":[113],"20,":[116],"5,":[119],"otherwise.":[122],"The":[123,167],"proof":[124],"gives":[125],"sharp":[127],"structural":[128],"guarantees":[129,164],"trees":[131],"most":[134],"pendant":[136],"paths":[137],"each":[139],"spine":[140,143],"vertex:":[141],"nineteen":[142],"vertices,":[144],"or":[145],"sixteen":[146],"vertices":[147],"bearing":[148],"arms":[150],"two,":[155],"suffice":[156],"colors.":[159],"Explicit":[160],"examples":[161],"show":[162],"both":[163],"are":[165],"sharp.":[166],"finite":[168],"computations":[169],"have":[170],"standalone":[171],"certificates;":[172],"unbounded":[174],"claims":[175],"follow":[176],"from":[177],"stated":[179],"constructions,":[180],"closure":[181],"arguments,":[182],"induction.":[184]},"counts_by_year":[],"updated_date":"2026-11-10T11:39:43.021295","created_date":"2026-11-09T00:00:00"}
