[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82436-en":3,"doc-seo-82436-105":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},82436,7971461741311,"Ophelia","https://ap-avatar.wpscdn.com/avatar/74000253aff267980c6?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779345379180704826",8,"Research & Report","Cut-homotopies and the Complexity of Edge-coloring Problems","Computational complexity of edge-coloring problems is studied for graphs where no edge-colored clique from a fixed finite family is allowed as an induced subgraph. For every such forbidden family, the corresponding decision problem is shown to be polynomial-time equivalent to a constraint satisfaction problem, producing a P vs. NP-complete dichotomy. The key step reduces CSP instances to edge-colorings using Ramsey-theoretic techniques and a new cut-homotopy notion that controls how color constraints propagate across graph structure.","CUT-HOMOTOPIES AND THE COMPLEXITY OF EDGE-COLORING PROBLEMS  \nALEXEY BARSUKOV, ROMAN FELLER, MAXIMILIAN HADEK, AND DAVIDE PERINTI  \nAbstract. We study the computational complexity of problems that ask if a given graph admits an edge-coloring that does not contain an edge-colored clique from some fixed finite family. We show that every such problem is poly-time equivalent to a Constraint Satisfaction Problem, yielding a P vs. NP-complete dichotomy. Our main contribution lies in the reduction from the CSP to the coloring problem where we apply methods from Ramsey theory and a novel notion of cut-homotopy.  \narXiv :2607 .09631v1 [math .CO] 10 Jul 2026  \n1. Introduction  \nFix a finite family F of edge-colored graphs and let A be the set of occurring colors. Let Col(F) be the following computational problem:  \nIs there an edge-coloring of a given input graph using colors from A so that no member of F appears as an induced edge-colored subgraph?  \nIf there is such an edge-coloring we refer to it as an F-free coloring. Clearly, every such problem Col(F) is contained in NP; moreover, a recent result due to Kun and Neˇsetˇril [43, Th. 4.1] asserts that the class of computational problems arising in this fashion is NP-rich. In search of subclasses avoiding NP-richness a natural next step is therefore to restrict the shape of forbidden graphs. In the present paper, we consider the case where every member of F is a colored clique. One of the earliest works in this direction dates back to Garey and Johnson’s list of NP-complete problems [34] which contains the problem Col(F), where F consists of monochromatic triangles in two colors. Later, this was extended to monochromatic cliques of arbitrary size ≥ 3 [30] and to arbitrarily many colors [2, 51] . We extend these hardness results to a P vs. NP-complete dichotomy for Col(F), when F consists of arbitrary edge-colored cliques.  \nTheorem 1.1 . For every finite family F of edge-colored cliques, the problem Col(F) is either solvable in polynomial time or NP-complete.  \nSuch coloring problems with forbidden cliques can be expressed in GMSNP, a logic capturing more general coloring problems. It was introduced independently in [8, 44] and has been studied extensively in subsequent work [1, 3 , 4 , 9 , 13 , 14 , 18 , 32 , 36] . Our result makes progress on the question whether GMSNP admits a complexity dichotomy, raised in [8] . It also makes progress on the Bodirsky–Pinsker conjecture [25] predicting a P vs. NP-complete dichotomy for an even more general class of forbidden-pattern coloring problems. Despite intense development of the theoretical  \nThe first three authors are supported by the European Unions ERC Synergy Grant 101071674, POCOCOP. Views and opinions expressed are however those of the authors only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them. Davide Perinti is supported by the Deutsche Forschungsgemeinschaft (DFG) under the project number 534904934 .  \nCUT-HOMOTOPIES AND THE COMPLEXITY OF EDGE-COLORING PROBLEMS 2  \nFigure 1 . Application of a cut-homotopy h between f, g to a tuple of colorings of K4 . The blue plane indicates the cut.  \ntools to tackle such problems [7, 10 , 21 , 22 , 24 , 26 , 33 , 45 , 47 , 48], see also [11], a proof of the general conjecture—while being verified for various subclasses [9, 12 , 13 , 15–17, 19 , 20 , 23 , 32 , 41 , 46 , 48]—still seems to be out of reach. One fundamental challenge lies in overcoming the problem of coloring with more than two “relevant” colors. This is exemplified by the fact that it proves to be a core challenge in a unified approach to the above dichotomies initiated in [48] . For further discussion of this challenge, we refer the reader to the concluding discussion in [32] .  \n1.1. An upper bound for the complexity. The constraint satisfaction problem associated toa relational str","cbCaidha1ZHF1nLF","https://ap.wps.com/l/cbCaidha1ZHF1nLF","pdf",433564,1,20,"English","en",105,"# Introduction\n## Problem Col(F) and F-free colorings\n## Upper bound via CSP reduction\n## Polymorphisms and cut-homotopies","[{\"question\":\"What does the problem Col(F) ask about graph edge-colorings?\",\"answer\":\"Given a finite family F of edge-colored graphs, Col(F) asks whether an input graph has an edge-coloring using available colors such that no member of F occurs as an induced edge-colored subgraph (an F-free coloring).\"},{\"question\":\"How are the edge-coloring problems related to constraint satisfaction problems?\",\"answer\":\"Each Col(F) can be reduced in polynomial time to a CSP on a structure A whose domain is the color set and whose relations encode exactly the F-free clique colorings. F-free colorings correspond precisely to homomorphisms from the constructed clique-encoding structure to A.\"},{\"question\":\"What is the main dichotomy result for Col(F) when forbidden graphs are edge-colored cliques?\",\"answer\":\"For any finite family F of edge-colored cliques, Col(F) is either solvable in polynomial time or is NP-complete, yielding a P vs. NP-complete dichotomy.\"}]",1784180372,50,{"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},"cut-homotopies-and-the-complexity-of-edge-coloring-problems","",{"@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/cut-homotopies-and-the-complexity-of-edge-coloring-problems/82436/",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 problem Col(F) ask about graph edge-colorings?","Question",{"text":75,"@type":76},"Given a finite family F of edge-colored graphs, Col(F) asks whether an input graph has an edge-coloring using available colors such that no member of F occurs as an induced edge-colored subgraph (an F-free coloring).","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How are the edge-coloring problems related to constraint satisfaction problems?",{"text":80,"@type":76},"Each Col(F) can be reduced in polynomial time to a CSP on a structure A whose domain is the color set and whose relations encode exactly the F-free clique colorings. F-free colorings correspond precisely to homomorphisms from the constructed clique-encoding structure to A.",{"name":82,"@type":73,"acceptedAnswer":83},"What is the main dichotomy result for Col(F) when forbidden graphs are edge-colored cliques?",{"text":84,"@type":76},"For any finite family F of edge-colored cliques, Col(F) is either solvable in polynomial time or is NP-complete, yielding a P vs. NP-complete dichotomy.","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},[93,97,101,105,110,114,119,122,126,129,133],{"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":28,"slug":113},6,"Technology","technology",{"id":115,"doc_module":4,"doc_module_name":45,"category_name":116,"show_sort_weight":117,"slug":118},7,"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":21,"slug":125},9,"Religion & Spirituality","religion-spirituality",{"id":21,"doc_module":4,"doc_module_name":45,"category_name":127,"show_sort_weight":21,"slug":128},"World Cup","world-cup",{"id":130,"doc_module":4,"doc_module_name":45,"category_name":131,"show_sort_weight":130,"slug":132},10,"Lifestyle","lifestyle",{"id":134,"doc_module":4,"doc_module_name":45,"category_name":135,"show_sort_weight":106,"slug":136},19,"General","general"]