[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82684-en":3,"doc-seo-82684-105":30,"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":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},82684,3848291630094,"Emma Wilson","https://eur-avatar.wpscdn.com/davatar_085a072bc5b1113ac321206ff7593b45",8,"Research & Report","Polynomial Algorithms for Minimum Degree Partitions in Semicomplete Digraphs","A 2-partition divides a digraph’s vertices into two nonempty parts with prescribed degree thresholds. Bang-Jensen and Christiansen conjectured that minimum-degree 2-partition problems become polynomial-time solvable on semicomplete digraphs when thresholds are fixed. This work resolves the conjecture by giving deterministic polynomial-time algorithms deciding the existence of three types of (δ+ , δ− , δ0)-constrained partitions and constructing a valid partition when it exists, using small degree certificates, minimal cores, closure and protective-set arguments, and deterministic universal colorings with monotone recoloring.","arXiv :2607 .026 19v 1 [ cs .DM] 2 Jul 2026  \nPolynomial Algorithms for Minimum Degree Partitions in  \nSemicomplete Digraphs  \nHanzhi Bai 1 Jin Yan 1 ∗  \nAbstract  \nA 2-partition of a digraph is a partition of its vertex set into two nonempty parts. Degreeconstrained 2-partition problems are generally computationally difficult, even when the prescribed properties are expressed only in terms of minimum indegree, minimum outdegree, or minimum semidegree. Bang-Jensen and Christiansen [2] conjectured that the minimumdegree partition problems would be polynomial-time solvable on semicomplete digraphs when the degree thresholds are fixed, and Bang-Jensen and Gutin [3] posed the related Problems 2 .8. 15 and 2 .8.16.  \nWe resolve this conjecture. More precisely, for every fixed pair of integers k 1 , k2 ≥ 2, we give deterministic polynomial-time algorithms that decide whether a given semicomplete digraph admits a (δ+ ≥ k 1 , δ − ≥ k2 )-partition, a (δ+ ≥ k 1 , δ0 ≥ k2 )-partition, or a (δ0 ≥ k 1 , δ0 ≥ k2 )-partition, and construct such a partition whenever one exists. Here, δ + , δ − , δ0 represent the minimum out-, in-, semi-degree, respectively. The algorithms use small degree certificates, minimal cores, closure and protective-set arguments, and deterministic universal colorings with monotone recoloring, which develop a new method in partition algorithm construction.  \nKeywords: semicomplete digraphs; 2-partitions; polynomial-time algorithms  \nMathematics Subject Classification: 05C20, 05C85, 68Q25 .  \n1 Introduction  \nA 2-partition of a graph or digraph G is a partition of its vertex set into two disjoint nonempty subsets V1 and V2 ; that is, V (G) = V1 ∪ V2 . Let P1 and P2 be two prescribed properties of graphs or digraphs. We say that G admits a (P1 , P2 )-partition if there exists a 2-partition (V1 , V2 ) of V (G) such that G [V1] satisfies P1 and G [V2] satisfies P2 . Typically, P1 and P2 may involve connectivity requirements, degree constraints, the existence or exclusion of certain subgraphs, or other structural conditions.  \nThe study of graph 2-partitions is an important topic in graph theory and has attracted considerable attention. Degree constraints have long played a central role in the study of graph partitions, scholars have made contributions in this field. The most classic result is the paper by Stiebitz in 1996 [16] . He proved that for positive integers s and t, every graph G with  \n1 School of Mathematics, Shandong University, Jinan 250100, P. R. China. Emails: [bhz@mail.sdu.edu.cn](bhz@mail.sdu.edu.cn) (H. Bai), Supported by the National Natural Science Foundation of China (Grant No. 12571373) and Natural Science Foundation of Shandong Province (Grant No. ZR2025MS05)  \n∗ Corresponding author. E-mail: [yanj@sdu.edu.cn](yanj@sdu.edu.cn) . (J.Yan)  \nthe minimum degree δ (G) ≥ s + t + 1 admits a (δ ≥ s, δ ≥ t)-partition. Recently, Wang and Wu [19] proved the average-degree partition theorem conjectured by Csóka, Lo, Norin, Wu, and Yepremyan, while Ma and Yu [13] settled a conjecture of Bollobás and Scott on sparse bipartitions. Further results on graph 2-partitions can be found in [6–8, 10–12, 18] .  \nFor digraphs, research on degree constrained partition problems is also meaningful. For adigraph D, we denote that δ + (D) = min v∈V(D) d+H(v), δ − (D) = min v∈V(D) d−H(v) and δ0 (D) = min{δ+ (D), δ − (D)} . In [1] by Alon in 1996 and [15] by Stiebitz in 1995, they ask, separately, whether for every pair of positive integers k 1 , k2 , there is a function f (k1 , k2 ) such that every digraph with minimum outdegree at least f (k1 , k2 ) admits a (δ+ ≥ k 1 , δ + ≥ k2 )-partition. Infact, as early as 1983, Thomassen [17] proved that every digraph D with δ + (D) ≥ 3 yields a (δ+ ≥ 1, δ+ ≥ 1)-partition, which means that f (1, 1) = 3 . But for larger k 1 , k2 , even the existence of f (1, 2) is still open. Recently, Steiner et al. [5] proved that if f (2, 2) exists, then all the numbers f (k1 , k2 ) with k 1 , k2 ≥ 1 exis","cbCaitbvanZ4QWMu","https://ap.wps.com/l/cbCaitbvanZ4QWMu","pdf",495894,4,1,15,"English","en",105,"# Introduction\n## Degree-constrained 2-partitions in graphs\n## Degree-constrained 2-partitions in digraphs\n## Semicomplete digraphs and the conjecture\n## Main result and theorem outline","[{\"question\":\"What problem does the paper study in semicomplete digraphs?\",\"answer\":\"It studies whether a semicomplete digraph admits a 2-partition of its vertex set where each part satisfies specified minimum outdegree, indegree, or semidegree constraints.\"},{\"question\":\"What main conjecture is resolved?\",\"answer\":\"Conjecture 1.1 by Bang-Jensen and Christiansen: for any fixed integers k1, k2 ≥ 2, the corresponding degree-threshold 2-partition decision problems are solvable in polynomial time on semicomplete digraphs.\"},{\"question\":\"What do the proposed algorithms guarantee?\",\"answer\":\"For each fixed pair k1, k2 ≥ 2, the algorithms deterministically decide whether a semicomplete digraph has the required (δ+ , δ− )-type, (δ+ , δ0)-type, or (δ0 , δ0)-type partition, and they construct such a partition when it exists.\"}]",1784182274,38,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"polynomial-algorithms-for-minimum-degree-partitions-in-semicomplete-digraphs","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":20},"https://docshare.wps.com/document/polynomial-algorithms-for-minimum-degree-partitions-in-semicomplete-digraphs/82684/",{"url":52,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-23","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 problem does the paper study in semicomplete digraphs?","Question",{"text":75,"@type":76},"It studies whether a semicomplete digraph admits a 2-partition of its vertex set where each part satisfies specified minimum outdegree, indegree, or semidegree constraints.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What main conjecture is resolved?",{"text":80,"@type":76},"Conjecture 1.1 by Bang-Jensen and Christiansen: for any fixed integers k1, k2 ≥ 2, the corresponding degree-threshold 2-partition decision problems are solvable in polynomial time on semicomplete digraphs.",{"name":82,"@type":73,"acceptedAnswer":83},"What do the proposed algorithms guarantee?",{"text":84,"@type":76},"For each fixed pair k1, k2 ≥ 2, the algorithms deterministically decide whether a semicomplete digraph has the required (δ+ , δ− )-type, (δ+ , δ0)-type, or (δ0 , δ0)-type partition, and they construct such a partition when it 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