[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83162-en":3,"doc-seo-83162-105":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},83162,2336464648746,"Skyler","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","Coloring Digraphs with b Colors","The dichromatic number of a digraph is defined as the minimum number of colors required to partition its vertices into acyclic subdigraphs. A biclique is a vertex set that induces all opposite-arc pairs. For a digraph D, the parameter Δ(D) is defined using the maximum product of in-degree and out-degree. The work proves that for any fixed b, all sufficiently large digraphs with Δ(D)=Δ either contain a biclique larger than Δ−2b or have dichromatic number at most Δ−b.","COLORING DIGRAPHS WITH  − b COLORS  \nKEN-ICHI KAWARABAYASHI AND LUCAS PICASARRI-ARRIETA  \nAbstract . The dichromatic number of a digraph is the minimum number of colors needed to partition its vertex set into acyclic subdigraphs. A bi c˜lique is a set of vertices inducing all  \npossible pairs of opposite arcs. For a digraph D, define ∆(D) = maxv∈V(D) pd+ (v) · d − (v) .  \nWe prove that, for every fixed integer b ∈ N, every digraph D with (D) = ∆ being sufficiently large with respect to b either contains a biclique whose size exceeds ∆ − 2b or has dichromatic number at most ∆ − b.  \nThis extends a classical result of Reed to the directed setting and supports a conjecture of the present authors. Further˜more, the theorem is tight, as for all integers b and ∆ ⩾ 3b  \nthere exists a digraph D with ∆(D) = ∆, dichromatic number ∆ − b + 1, and whose largest biclique has size ∆ − 2b + 1 .  \narXiv :2607 .06928v1 [math .CO] 8 Jul 2026  \nResearch supported by JSPS KAKENHI JP20A402 and 22H05001 and by JST ASPIRE JPMJAP2302 .  \n2 K. KAWARABAYASHI AND L. PICASARRI-ARRIETA  \n1. Introduction  \nIt is well-known that the chromatic number χ(G) of a graph G is bounded above by ∆(G)+1 , where ∆(G) denotes the maximum degree of G. This bound is achieved by complete graphs, which motivates the study of the relationships between χ (G) , ∆(G), and the clique number ω (G) of a graph G. As a first step in this direction, Brooks [9] obtained a seminal result stating that every graph G satisfies  \nχ (G) ⩽ max{∆(G),ω (G)} ,  \nunless ∆(G) = 2 and G contains an odd cycle. Pushing this direction further, in 1977 Borodin and Kostochka [8] posed a celebrated conjecture stating that  \nχ (G) ⩽ max{∆(G) − 1,ω(G)}  \nholds for every graph G with ∆(G) ⩾ 9. Even though the conjecture is still open in general, Reed [32] proved that it holds for graphs G with ∆(G) ⩾ 1010. Moreover, in 1998 Reed [31] conjectured that these inequalities are just the tip of the iceberg, and posed the following famous conjecture.  \nConjecture 1 ([31]) . Every graph G satisfies χ (G) ⩽ ⌈ ~~1~~2 (∆(G) + 1 + ω(G))⌉ .  \nAs supporting evidence for the conjecture, in the same paper Reed proved that, for every graph G, χ (G) ⩽ ~~1~~2 􀀀∆(G) + 1 + ω(G)􀀁 holds whenever ∆(G) is sufficiently large and ω (G) is sufficiently close to ∆(G); formally when ∆(G) ⩾ ∆0 and ω (G) ⩾ (1 − ε)∆(G) for some absolute constants ∆0 ∈ N and ε ∈ (0 , 1) . This has the following two main consequences. Corollary 2 ([31]) . There exists ε > 0 such that every graph G satisfies  \nχ (G) ⩽ ⌈(1 − ε)(∆(G) + 1) + εω(G)⌉ .  \nCorollary 3 ([31]) . For every b ∈ Z, there exists ∆ b ∈ N such that every graph G with ∆(G) ⩾ ∆b and ω(G) ⩽ ∆(G) − 2b satisfies χ(G) ⩽ ∆(G) − b.  \nObserve that Conjecture 1 is precisely the statement of Corollary 2 for ε = 1/2 . This gave rise to a line of research aimed at determining the largest value of ε > 0 for which Corollary 2 holds. The current best result is due to Hurley, Joannis de Verclos, and Kang [18], who proved that it holds for ε = 0 .119 for graphs with sufficiently large maximum degree. This improveson earlier results obtained by Bonamy, Perrett, and Postle [7] and by Delcourt and Postle [10] .  \nLet us briefly discuss the sharpness of Corollary 3 and its connection to Conjecture 1. First, observe that the threshold ∆ b in Corollary 3 must depend on b. Indeed, for every integer b ∈ N, there exists a graph G with ∆(G) = 6b + 2 , ω (G) = 4b + 2, and χ (G) = 5b + 3 . One example is the graph obtained from a 5-cycle by blowing up each vertex into a clique of size 2b + 1 . In particular, such a graph does not satisfy the conclusion of Corollary 3, showing that ∆ b must be at least 6b + 3 .  \nHowever, the dependence of ∆b on b might not be necessary under the stronger hypothesis ω (G) ⩽ ∆ − 2b − 1. Indeed, Conjecture 1 is easily seen to be equivalent to the following. Conjecture 4 . For every b ∈ Z and every graph G, if ω(G) ⩽ ∆ − 2b − 1 then χ(G) ⩽ ∆ − b.  \nUnder the assumption that ∆ b is arbitra","cbCaiq6P4uvhs8v9","https://ap.wps.com/l/cbCaiq6P4uvhs8v9","pdf",569347,4,1,30,"English","en",105,"# Abstract\n# Introduction\n## Background on chromatic number and degree bounds\n## Reed and related conjectures\n## Motivation for a directed extension\n# Directed digraph parameters\n## Definitions of Δ(D), biclique number, and dichromatic number\n## Known directed analogues of Brooks-type results\n# Main theorem direction and tightness","[{\"question\":\"What does the dichromatic number measure in this work?\",\"answer\":\"It measures the smallest number of colors needed so the digraph’s vertex set can be partitioned into acyclic subdigraphs.\"},{\"question\":\"How are bicliques and the biclique number defined for digraphs?\",\"answer\":\"A biclique is a vertex set inducing all possible arcs of the appropriate complete directed form, and the biclique number is the size of the largest such biclique.\"},{\"question\":\"What is the main result regarding bicliques and dichromatic number for fixed b?\",\"answer\":\"For each fixed integer b, sufficiently large digraphs with a given Δ either contain a biclique of size exceeding Δ−2b or have dichromatic number at most Δ−b, and the statement is shown to be tight.\"}]",1784185686,76,{"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},"coloring-digraphs-with-b-colors","",{"@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/coloring-digraphs-with-b-colors/83162/",{"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","2026-07-24","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 does the dichromatic number measure in this work?","Question",{"text":75,"@type":76},"It measures the smallest number of colors needed so the digraph’s vertex set can be partitioned into acyclic subdigraphs.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How are bicliques and the biclique number defined for digraphs?",{"text":80,"@type":76},"A biclique is a vertex set inducing all possible arcs of the appropriate complete directed form, and the biclique number is the size of the largest such biclique.",{"name":82,"@type":73,"acceptedAnswer":83},"What is the main result regarding bicliques and dichromatic number for fixed b?",{"text":84,"@type":76},"For each fixed integer b, sufficiently large digraphs with a given Δ either contain a biclique of size exceeding Δ−2b or have dichromatic number at most Δ−b, and the statement is shown to be 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