[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83019-en":3,"doc-seo-83019-105":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},83019,7971461740909,"Levi","https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d",8,"Research & Report","Coloring T-Perfect Graphs With Fewer Colors","Coloring t-perfect graphs is addressed by improving the best known finite bound on their chromatic number. Prior work proved t-perfect graphs are 199053-colorable using an arithmetic-rope structure that forces an odd wheel as a t-minor. The refined argument guarantees an arithmetic rope under only a linear chromatic-number assumption, then further weakens the structure to quasi-arithmetic ropes, enabling an even smaller bound: every t-perfect graph is 186-colorable and, consequently, every h-perfect graph is (ω(G)+184)-colorable.","arXiv :2607 .05961v1 [math .CO] 7 Jul 2026  \nColoring t-perfect graphs with fewer colors  \nMatija Novakovi´c Stefan Weltge  \nJuly 8, 2026  \nAbstract  \nRecently, Chudnovsky, Cook, Davies, Oum, and Tan obtained the first finite bound on the chromatic number of t-perfect graphs, showing that they are 199053-colorable. We improve this bound to 186 by refining their proof.  \nThe original proof establishes that every graph with large odd girth and large chromatic number contains a certain structure called an r-arithmetic rope, and that its existence in a certain leveling of a graph with large odd girth would imply an odd wheel as a t-minor, a known obstruction of t-perfectness. While their technique requires a lower bound on the chromatic number that is exponential in r, we show that the existence of an r-arithmetic rope can already be guaranteed under a linear bound. Using a slightly weakened notion of arithmetic ropes allows us to reduce the bound even further.  \n1 Introduction  \nThe stable set problem asks for finding a maximum stable set in a given undirected graph, i.e., a largest set of pairwise non-adjacent vertices. While the problem is NP-hard in general, it can be solved in polynomial time for several classes of graphs. A common approach to derive efficient algorithms is to exploit the structure of the stable set polytope, which is the convex hull of characteristic vectors of stable sets. Introduced by Chv´atal [5], a prominent example is the class of t-perfect graphs, which are defined as the graphs G whose stable set polytope coincides with the set of vectors x ∈ [0 , 1] V (G) satisfying xu + xv ≤ 1 for every edge uv ∈ E (G) (edge inequalities) and Pv∈V(C) xv ≤ ~~1~~2 (|V(C)| − 1) for every odd cycle C of G (odd cycle inequalities) .  \nIn 1992, Shepherd [7, 8.14] asked whether the stable set polytope P of a t-perfect graph G always has the integer decomposition property, that is, for every positive integer k, every integer point in kP is a sum of k integer vectors points in P. Since the all-~~1~~3 vector is contained P, this would imply that G is 3-colorable. In 1994, Laurent and Seymour [9, p. 1207](and later Benchetrit [1, 2]) found t-perfect graphs with chromatic number 4, negatively answering Shepherd’s question. Since then, it is an open question whether every t-perfect graph is 4-colorable [7, 8.14] .  \nOnly recently, Chudnovsky, Cook, Davies, Oum, and Tan [4] gave the first finite bound on the chromatic number oft-perfect graphs, showing that they are 199053-colorable. While their proof has not been optimized for the best bound, they expected that carrying out their arguments with more care  \n“[...] would still leave a large gap between our bound and the lower bound of 4 . A good milestone for narrowing the gap would be to improve our upper bound to at most 1000 .”  \nWe show how their proof can be refined to obtain the following bound.  \nTheorem 1.1 . Every t-perfect graph is 186-colorable.  \nThe approach of [4] is based on only two properties of t-perfect graphs: they do not contain odd wheels as t-minors, and they have a stable set that intersects every shortest odd cycle. As the main technical contribution, the authors show that every graph with large odd girth and large chromatic number contains a certain structure called an ”arithmetic rope”, and observe that the existence of an arithmetic rope in a  \ncertain leveling of a graph with large odd girth would imply the existence of an odd wheel as a t-minor. Werefine their proof and show that the existence of an arithmetic rope can be guaranteed under a much weaker assumption on the chromatic number. To decrease the bound on the chromatic number even further, we exploit the fact that a slightly weaker structure, which we call a quasi-arithmetic rope, is already sufficient for finding an odd wheel t-minor.  \nTheorem 1.1 also yields an improved bound on the chromatic number of h-perfect graphs, which are defined as the graphs G whose stable set polytope coincides w","cbCaimLwANSnfvO1","https://ap.wps.com/l/cbCaimLwANSnfvO1","pdf",737520,1,15,"English","en",105,"# Introduction\n## Stable set polytope and t-perfect graphs\n## Prior bounds and motivation for improvement\n## Main results and implications for h-perfect graphs\n# Quasi-arithmetic ropes and t-minors of odd wheels\n## Closure under t-minors and t-contractions\n## Odd wheels and odd girth reduction","[{\"question\":\"What does the paper prove about the chromatic number of t-perfect graphs?\",\"answer\":\"It proves that every t-perfect graph is 186-colorable, improving the previous finite bound of 199053-colorable obtained by earlier work.\"},{\"question\":\"How does the proof connect arithmetic ropes to t-minors of odd wheels?\",\"answer\":\"The approach shows that a graph with large odd girth and large chromatic number contains an arithmetic rope, and the existence of such a rope (in an appropriate leveling) implies an odd wheel as a t-minor.\"},{\"question\":\"What is the relationship between the results for t-perfect and h-perfect graphs?\",\"answer\":\"Using a known implication between colorability of t-perfect and h-perfect graphs, the paper derives a corollary that every h-perfect graph is (ω(G)+184)-colorable.\"}]",1784184701,38,{"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},"coloring-t-perfect-graphs-with-fewer-colors","",{"@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/coloring-t-perfect-graphs-with-fewer-colors/83019/",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 does the paper prove about the chromatic number of t-perfect graphs?","Question",{"text":75,"@type":76},"It proves that every t-perfect graph is 186-colorable, improving the previous finite bound of 199053-colorable obtained by earlier work.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the proof connect arithmetic ropes to t-minors of odd wheels?",{"text":80,"@type":76},"The approach shows that a graph with large odd girth and large chromatic number contains an arithmetic rope, and the existence of such a rope (in an appropriate leveling) implies an odd wheel as a t-minor.",{"name":82,"@type":73,"acceptedAnswer":83},"What is the relationship between the results for t-perfect and h-perfect graphs?",{"text":84,"@type":76},"Using a known implication between colorability of t-perfect and h-perfect graphs, the paper derives a corollary that every h-perfect graph is 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