[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85371-en":3,"doc-seo-85371-105":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},85371,7971461741311,"Ophelia","https://ap-avatar.wpscdn.com/avatar/74000253aff267980c6?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779345379180704826",8,"Research & Report","Decision Problem for Hamilton 2-Cycles in 4-Graphs","Decision problem for Hamilton 2-cycles in 4-uniform hypergraphs: a cyclic ordering of 2t vertices forms a Hamilton 2-cycle when each consecutive four-tuple defines an edge modulo 2t. For every γ>0 and sufficiently large n, the work characterizes n-vertex 4-uniform hypergraphs where every vertex triple lies in at least (1/3+γ)n edges and guarantees a Hamilton 2-cycle. The bound is asymptotically tight up to γn and confirms a conjecture of Garbe and Mycroft, yielding a polynomial-time decision algorithm.","DECISION PROBLEM FOR HAMILTON 2-CYCLES IN 4-GRAPHS  \nLUYINING GAN, JIE HAN, AND BIN WANG  \nAbstract . A 4-uniform 2-cycle in a 4-uniform hypergraph of length t is a cyclic ordering of 2t vertices v 1 v2 ··· v2tv1 such that v2i+1v2i+2v2i+3v2i+4 are edges for 0 ≤ i ≤ t − 1 while the addition is modulo 2t.  \nFor every γ > 0 and large n, we characterize the n-vertex 4-uniform hypergraphs such that every triple of vertices is contained in at least (1/3+γ)n edges and admits a Hamilton 2-cycle. Up to the error term γn, the assumption on the minimum codegree is best possible and verifies a conjecture of Garbe and Mycroft. As a consequence, this gives a polynomial-time algorithm that decides whether an n-vertex 4-uniform hypergraph with minimum codegree (1/3+γ)n contains a Hamilton 2-cycle. This stands as a steep contrast to the graph case where such a hardness gap has size o (n) .  \narXiv :2607 . 11872v1 [math .CO] 13 Jul 2026  \n1. Introduction  \nThe Hamilton cycle problem looks for a spanning cycle of a given graph. It is one of the original 21 NP-complete problems by Karp [21] . The study of Hamiltonicity has been one of the central topics in graph theory and computer science [3, 5], due to its broad applications and connections with real-world problems, such as TSP, routing, and scheduling [21, 29, 12] . Given its importance, it is desirable to extend the study of Hamilton cycles to hypergraphs, which have strong potential for modeling more sophisticated multi-object relations in applications.  \nA k-uniform graph P is an ℓ-path if P has no isolated vertices and moreover, the vertices of P can be ordered in such a way that every edge of P consists of k consecutive vertices and every two consecutive edges intersect in precisely ℓ vertices. Furthermore, if the ordering is cyclic, then P is called an ℓ-cycle. Naturally, we say that an n-vertex k-graph contains a Hamilton ℓ-cycle if thereis a subgraph of H which is a copy of ℓ-cycle covering all vertices of H. Note that it is necessary that (k − ℓ) divides n. The length of an ℓ-path or ℓ-cycle is the number of edges it contains. When ℓ = k − 1, it is also referred to as a tight path or tight cycle, which is considered in general as the most interesting case – one simple reason is that it contains other ℓ-paths or cycles as subgraphs.  \nLet H be an n-vertex k-uniform graph (k-graph for short) . The degree of a set S of vertices is the number of edges containing S. Denote by δ d(H) its minimum codegree, that is, the minimum of the degree of S (denoted by deg(S)) over all S of size d where d ∈ [k − 1] . We refer to δ k−1(H) as the minimum codegree of H. There has been a strong focus on the study of minimum (co)degree conditions forcing the existence of Hamilton ℓ-cycles in k-graphs, pioneered by the celebrated work of Rödl, Ruciński and Szemerédi [37] on an approximate version of the Dirac theorem in k-graphs, as well as the absorbing method developed by them in this thread of research. In particular, in [37] it is shown that a minimum codegree (1/2 + o(1))|H| guarantees the existence of a tight Hamilton cycle in a k-graph H.  \n1.1. Algorithmic problems – below-guarantee. Beyond sufficient conditions, the below-guarantee problem was first investigated by Garbe and Mycroft [11] . For tight Hamilton cycles, they showed a strong hardness result saying that for k ≥ 3, there exists a constant C such that the decision  \nproblem for tight Hamilton cycles in n-vertex k-graphs with minimum codegree n/2 − C is NPcomplete. That is, in terms of minimum codegree assumptions, the gap between intractable and trivially tractable is o (n) .  \nOn the other hand, the main result of Garbe and Mycroft [11] shows that for 4-uniform 2-cycles, the corresponding gap is as large as Θ(n) . That is, they showed that there exists ε > 0, such that for every n-vertex 4-graph H with δ3 (H) ≥ ~~n~~2 −εn the Hamilton 2-cycle problem is tractable. Moreover, they achieve it by characterizing all such 4-graphs H with Hamilton","cbCaivpjfd1pUJ9h","https://ap.wps.com/l/cbCaivpjfd1pUJ9h","pdf",732905,1,43,"English","en",105,"# Introduction\n## Algorithmic problems – below-guarantee\n## The decision problem HAM(k,ℓ,δ)","[{\"question\":\"What is a Hamilton 2-cycle in a 4-uniform hypergraph?\",\"answer\":\"It is a cyclic ordering of 2t vertices such that each consecutive set of four vertices forms an edge, with indices taken modulo 2t.\"},{\"question\":\"What minimum codegree condition guarantees a Hamilton 2-cycle in the main result?\",\"answer\":\"For every γ\\u003e0 and large n, if every triple of vertices is contained in at least (1/3+γ)n edges, then the hypergraph admits a Hamilton 2-cycle.\"},{\"question\":\"What algorithmic consequence follows from the characterization?\",\"answer\":\"A polynomial-time algorithm decides whether an n-vertex 4-uniform hypergraph with minimum codegree (1/3+γ)n contains a Hamilton 2-cycle.\"}]",1784202907,108,{"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},"decision-problem-for-hamilton-2-cycles-in-4-graphs","",{"@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/decision-problem-for-hamilton-2-cycles-in-4-graphs/85371/",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 a Hamilton 2-cycle in a 4-uniform hypergraph?","Question",{"text":75,"@type":76},"It is a cyclic ordering of 2t vertices such that each consecutive set of four vertices forms an edge, with indices taken modulo 2t.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What minimum codegree condition guarantees a Hamilton 2-cycle in the main result?",{"text":80,"@type":76},"For every γ>0 and large n, if every triple of vertices is contained in at least (1/3+γ)n edges, then the hypergraph admits a Hamilton 2-cycle.",{"name":82,"@type":73,"acceptedAnswer":83},"What algorithmic consequence follows from the characterization?",{"text":84,"@type":76},"A polynomial-time algorithm decides whether an n-vertex 4-uniform hypergraph with minimum codegree (1/3+γ)n contains a Hamilton 2-cycle.","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},[93,97,101,105,110,115,120,123,128,131,135],{"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":116,"doc_module":4,"doc_module_name":45,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":45,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":45,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]