[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84060-en":3,"doc-seo-84060-105":29,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},84060,1374391975076,"Riley","https://ap-avatar.wpscdn.com/avatar/14000253ca4ec9f6853?x-image-process=image/resize,m_fixed,w_180,h_180&k=1783305029341752051",8,"Research & Report","Star Coloring of Hypergraphs","Study focuses on a generalization of star coloring for hypergraphs. For a fixed family F of connected subhypergraphs, an F-coloring assigns vertices colors so that no hyperedge is monochromatic and no member of F is bicolored. For an r-uniform hypergraph with maximum degree d, the work defines χ_r^F(d) as the worst-case minimum number of colors over all such hypergraphs. Bounds are proved for χ_r^F(d), yielding consequences for star and acyclic coloring, including new asymptotic lower behavior via a balanced-triangle condition.","STAR COLORING OF HYPERGRAPHS  \nLALE ¨OZKAHYA AND POLAT SARIYERL˙I  \nAbstract. We study a generalization of the star coloring problem on hypergraphs. For a family of connected subhypergraphs F, we define an F-coloring of a hypergraph as a coloring avoiding monochromatic hyperedges and any 2-colored member ofF. We let χrF(d) be the maximum of the minimum number of colors needed for an F-coloring of an r-uniform hypergraph with maximum degree d. We show bounds for χrF(d), that also yield results on star and acyclic coloring problem on hypergraphs.  \narXiv :2607 .06082v1 [math .CO] 7 Jul 2026  \n1. Introduction  \nHypergraph coloring is widely studied, such as in [3, 4 , 5 , 6 , 12 , 11 , 13 , 19 , 17 , 20] . A hypergraph is called r-uniform if all its hyperedges have exactly r vertices. Erd˝os and Lov´asz [7] proved that the vertex set of any (r + 1)-uniform hypergraph with maximum degree ∆ can be colored with c∆1/r, for some constant c, such that there is no monochromatic hyperedge. This result is proved by an application of Lov´asz Local Lemma, which is also introduced in the same paper. In this paper, we aim to study a generalization of star coloring problem on hypergraphs. The star-coloring of a graph, introduced by Gr¨unbaum [15], is defined as the proper vertex coloring, where only stars are allowed to be bicolored (2-colored) . In the same paper, acyclic coloring of graphs is introduced, defined as a proper vertex coloring without bicolored cycles. The star and acyclic coloring problems are shown to be NP-complete by Albertson et al. [1] and Kostochka [18], respectively.  \nWe consider a generalization of these colorings to hypergraphs, by letting F be a fixed family of connected subhypergraphs and F-coloring of a hypergraph be a vertex coloring with no monochromatic hyperedge and no bicolored member of F. Let Krn denote the complete r-uniform hypergraphs on n vertices and ∆(H) denote the maximum vertex degree in a hypergraph H. For ahypergraph H, we let χF (H) be the minimum number of colors needed in an F-coloring of H and χrF(d) := max{χF (H) : H ⊂ Krn and ∆(H) = d} . We prove the following general upper bound on F-coloring of hypergraphs.  \nTheorem 1 . For r ≥ 2, let F be a family of connnected hypergraphs, 2 ≤ m = maxF∈F |F| , r + 1 ≤ a = minF∈F |V(F)| and t = min{m, a − 1} . Then χrF(d) ≤ ⌈(23a−2mt (rd)m ) ~~ ~~a~~ ~~2 ⌉ .  \nLet us call a triangle with edges E1 , E2 , E3 such that ∩3i=1 Ei = ∅ and |E1 ∩ Ej | = ⌊r/2⌋ for j = 2 , 3 , and |E2 ∩ E3 | = ⌈r/2⌉, a balanced triangle. For F-coloring, we prove the following result. This result provides a lower bound for star and acyclic colorings of hypergraphs as discussed in Section 2.  \nTheorem 2 . Let F be a family of connected r-uniform hypergraphs, also having the balanced triangle  \nas a member. Then, for r ≥ 2, χrF(d) = Ω((d3 /lnd) 3r4 ) .  \nKey words and [phrases.](phrases. star)[ star](phrases. star) coloring, acyclic coloring, hypergraphs, bicolored subgraph.  \n1  \nSTAR COLORING OF HYPERGRAPHS 2  \nIn Section 2, we discuss the implications of Theorems 1 and 2 on star and acyclic coloring of hypergraphs. In Section 3, we provide proofs of these theorems.  \n2. Star and Acyclic Coloring of Hypergraphs  \nA star hypergraph is a set of edges E1 , ... , Em , m ≥ 2, such that ∩iEi  ∅ . We call a subhypergraph with hyperedges E0 ,..., Em−1 a cycle of length m if Ei ∩Ei+1  ∅ and Ei−1 ∩Ei ∩Ei+1 = ∅ for each i modulo m. We let a star (acyclic, resp.) coloring of a hypergraph H be defined as a coloring with no monochromatic hyperedge, where only bicolored subhypergraphs are allowed to be stars (cycles of length at least three, resp.) . We let the star (acyclic, resp.) chromatic number of H,χs (H) (χa (H), resp.) be the minimum number of colors needed in a star (acyclic, resp.) coloring of H. We let χrs(d) = max{χs (H) : H ⊂ Krn and ∆(H) = d} and χra(d) = max{χa (H) : H ⊂ Krn and ∆(H) = d} By our definition, χra(d) ≤ χrs(d) .  \nFor graphs, the star coloring problem corresponds to an ","cbCaiaETS0WTgVlS","https://ap.wps.com/l/cbCaiaETS0WTgVlS","pdf",309417,1,7,"English","en",105,"# Introduction\n# Star and Acyclic Coloring of Hypergraphs\n## Implications of main theorems\n# Proofs","[{\"question\":\"What is an F-coloring of a hypergraph in this work?\",\"answer\":\"Given a family F of connected subhypergraphs, an F-coloring is a vertex coloring that avoids monochromatic hyperedges and also avoids any bicolored member of F.\"},{\"question\":\"How is χ_r^F(d) defined?\",\"answer\":\"χ_r^F(d) is the maximum, over r-uniform hypergraphs with maximum degree d, of the minimum number of colors required for an F-coloring.\"},{\"question\":\"How do the results relate to star coloring and acyclic coloring?\",\"answer\":\"The derived bounds on χ_r^F(d) imply corresponding upper and lower bounds for star and acyclic coloring problems on hypergraphs.\"}]",1784192296,18,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"star-coloring-of-hypergraphs","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/star-coloring-of-hypergraphs/84060/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What is an F-coloring of a hypergraph in this work?","Question",{"text":75,"@type":76},"Given a family F of connected subhypergraphs, an F-coloring is a vertex coloring that avoids monochromatic hyperedges and also avoids any bicolored member of F.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How is χ_r^F(d) defined?",{"text":80,"@type":76},"χ_r^F(d) is the maximum, over r-uniform hypergraphs with maximum degree d, of the minimum number of colors required for an F-coloring.",{"name":82,"@type":73,"acceptedAnswer":83},"How do the results relate to star coloring and acyclic coloring?",{"text":84,"@type":76},"The derived bounds on χ_r^F(d) imply corresponding upper and lower bounds for star and acyclic coloring problems on hypergraphs.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,119,122,127,130,134],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":45,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":45,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":21,"doc_module":4,"doc_module_name":45,"category_name":116,"show_sort_weight":117,"slug":118},"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":45,"category_name":124,"show_sort_weight":125,"slug":126},9,"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":45,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":45,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":45,"category_name":136,"show_sort_weight":106,"slug":137},19,"General","general"]