[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83635-en":3,"doc-seo-83635-105":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},83635,1649267921044,"Ava Thompson","https://us-avatar.wpscdn.com/avatar/1800007509477c92dfb?_k=1782875107921204101",8,"Research & Report","3-Colouring Graphs Excluding a Fixed Minor","For every fixed graph H, every n-vertex graph G that excludes H as a minor admits a 3-colouring with bounded clustering: each monochromatic component has size at most f(H)·n^(4/9). This result extends a recent improvement from planar graphs to all proper minor-closed graph classes. It strengthens the best known clustered-3-colouring upper bounds beyond the earlier O(sqrt(n)) estimate and provides a uniform clustering exponent independent of H, while aligning with known lower-bound behavior.","arXiv :2607 .02159v1 [math .CO] 2 Jul 2026  \n3-Colouring Graphs Excluding a Fixed Minor  \nVida Dujmovi∗ Hussein Houdrouge† Pat Morin†  \nAbstract  \nWe show that, for every fixed graph H, every n-vertex graph G that excludes H as a minor is 3-colourable with clustering OH(n4/9) . That is, there exists a function f such that for every graph H, every n ≥ 1, every n-vertex graph G that excludes H as a minor has a vertex colouring with 3 colours in which each monochromatic component has size at most f (H ) · n4/9 . This generalizes a recent result of Dujmovi, Morin, Norin, and Wood (arXiv:2507.03163) from planar graphs to all proper minor-closed graph classes and is the first improvement on clustered 3-colouring of proper minor-closed graph classes since the upper bound of OH( √n) due to Linial, Matouek, Sheffet, and Tardos (Comb. Prob. Comput., 17(4):577–589, 2008) .  \n1. Introduction  \nA vertex colouring of a graph G is a function ϕ : V ( G) → K where the elements of the set K are called colours. A vertex colouring is proper if ϕ (u ) , ϕ (v) for every edge uv of G. For k ∈ N, we say a graph G is k-colourable if there exists a proper colouring ϕ : V ( G) → K where the cardinality of K is at most k. Proper colouring is a widely studied subject within graph theory. The celebrated Four Colour Theorem for planar graphs asserts that every planar graph is 4-colourable [3, 39] . A major open problem is the Hadwiger Conjecture, which asserts that every graph that does not contain the complete graph K t on t vertices as a minor can be properly coloured with t − 1 colours.  \nIn the current paper, we study a relaxation of proper colouring. Given a (not necessarily proper) colouring ϕ of a graph G, a cluster is a connected subgraph of G whose vertices are all assigned to the same colour. A vertex-maximal cluster is called a monochromatic component. We say that a colouring of a graph G has clustering c if the size of every cluster is at most c. Thus, a proper colouring is a colouring with clustering one. Recently, clustered colouring has received considerable attention, as a way of approaching difficult conjectures on proper colouring (such as the Hadwiger Conjecture or Hajs Conjecture [8, 13, 15, 21, 24, 25, 33]), asan interesting question in its own right (see [4, 5, 10, 11, 17, 20, 22, 23, 27, 29–32, 34–38]), and for its application in databases as in [26] . For further details, we refer the reader to the survey by Wood [42] .  \nOne line of research considered by Linial et al. [27] asks about colouring planar graphs with two colours or three colours, rather than four. The authors show that planar graphs have 2-colourings with clustering O (n2/3) (which follows quickly from the Planar Separator Theorem [28]) and show that this is tight: there are n-vertex planar graphs in which every 2-  \n∗ School of Computer Science and Electrical Engineering, University of Ottawa.†School of Computer Science, Carleton University. Research partially funded by NSERC.  \ncolouring has a cluster of size Ω (n2/3) . The same authors show that n-vertex planar graphs (and, more generally, all proper-minor-closed families of graphs) have 3-colourings with clustering O (n 1/2) (which again follows quickly from the Planar Separator Theorem [28] and its extension to proper minor-closed graph classes [1]) . On the lower bound side, there are some n-vertex planar graphs in which every 3-colouring has a cluster of size Ω (n 1/3) . These two natural-looking bounds stood for many years, until recently, when Dujmovi et al. [11] showed that planar graphs have 3-colourings with clustering O (n4/9):  \nTheorem 1 (Dujmovi et al. [11]). Every n-vertex planar graph has a 3-colouring with clustering O (n4/9).  \nIn this paper, we consider a natural follow-up question: can Theorem 1 be generalized to graphs from any proper minor-closed family of graphs? We answer this question positively by successive generalization of this result. We first show that it can be generalized to all graphs","cbCaimaWFtoSiRP7","https://ap.wps.com/l/cbCaimaWFtoSiRP7","pdf",292809,4,1,19,"English","en",105,"# Introduction\n## Clustered colouring and minors\n## Main results (Theorems 1–3)\n## Proof overview","[{\"question\":\"What does it mean for a colouring to have clustering O(n^(4/9))?\",\"answer\":\"A colouring has clustering c if every monochromatic connected component has size at most c. Here, c is bounded by a constant depending on H times n^(4/9).\"},{\"question\":\"What is the main theorem about H-minor-free graphs?\",\"answer\":\"For any fixed graph H, every n-vertex graph G that excludes H as a minor has a 3-colouring whose monochromatic components have size at most O_H(n^(4/9)).\"},{\"question\":\"How does the paper generalize earlier planar-graph results?\",\"answer\":\"It starts from a planar-graph clustered 3-colouring bound of O(n^(4/9)) and then extends the guarantee from planar graphs to broader proper minor-closed classes: first bounded genus, then all graphs excluding a fixed minor.\"}]",1784189413,48,{"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},"3-colouring-graphs-excluding-a-fixed-minor","",{"@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/3-colouring-graphs-excluding-a-fixed-minor/83635/",{"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 does it mean for a colouring to have clustering O(n^(4/9))?","Question",{"text":75,"@type":76},"A colouring has clustering c if every monochromatic connected component has size at most c. Here, c is bounded by a constant depending on H times n^(4/9).","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What is the main theorem about H-minor-free graphs?",{"text":80,"@type":76},"For any fixed graph H, every n-vertex graph G that excludes H as a minor has a 3-colouring whose monochromatic components have size at most O_H(n^(4/9)).",{"name":82,"@type":73,"acceptedAnswer":83},"How does the paper generalize earlier planar-graph results?",{"text":84,"@type":76},"It starts from a planar-graph clustered 3-colouring bound of O(n^(4/9)) and then extends the guarantee from planar graphs to broader proper minor-closed classes: first bounded genus, then all graphs excluding a fixed minor.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":22,"doc_module":4,"doc_module_name":46,"category_name":136,"show_sort_weight":106,"slug":137},"General","general"]