[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-151274-en":3,"doc-seo-151274-105":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":4,"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":27,"seo_description":14,"update_tm":28,"read_time":29},151274,687207022233,"Connor ","https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d",8,"Research & Report","Random Sequential Nearest-Neighbor Coloring on Trees","We analyze a nearest-neighbor coloring process on graphs where vertices are revealed in random order and each inherits the color of the closest previously revealed vertex. The work studies regular trees and the genealogy formed by color inheritance, showing that unlike Euclidean settings, the genealogical graph on an infinite regular tree is disconnected. It has infinitely many infinite one-ended components with distinct asymptotic directions, while each vertex has finitely many descendants. The paper also examines modifications with finitely many initial seeds and local limits for growing finite trees under two seed configurations.","Random sequential nearest-neighbor coloring on trees  \nAnne-Laure Basdevant∗, Arvind Singh†  \nJune 24, 2026  \narXiv :2606 .24793v1 [math .PR] 23 Jun 2026  \nFigure 1: Simulation of the coloring process on a binary tree of height 15. All the leaves are seeds initially colored in red while the root is the seed of the blue cluster.  \nAbstract  \nWe study a nearest-neighbor coloring process in which vertices are revealed in random order and inherit the color of the closest vertex revealed before them. This model is a discrete analogue of coloring processes previously studied by Preater [10] and Aldous [1] in Euclidean spaces. We focus here on regular trees and analyze the associated genealogy of color inheritance. In contrast with the Euclidean case, the genealogical graph on an infinite regular tree is not connected: it has infinitely many infinite one-ended components, each with a distinct asymptotic direction, while every vertex has only finitely many descendants. We also describe how this structure is modified in the presence of finitely many initial seeds. Finally, we study local limits of the coloring on finite regular trees as their height tends to infinity, for two natural seed configurations: two fixed seeds, and one blue seed at the root with red seeds at the leaves.  \n1 Introduction  \nWe study a random coloring procedure on graphs, in which vertices are colored sequentially at random, each vertex taking, at the time it is picked, the color of one of its closest already colored vertices. This model can be interpreted as a growth process driven by local competition between different species (i. e . , colors) . Alternatively, it may also be interpreted as a variant of the classical voter model where new settlers take the political opinion of one of their already settled neighbors.  \nSequential nearest-neighbor rules of this kind have appeared in two closely related forms. In the uncolored version, each arriving point is linked to its closest predecessor, giving the on-line nearestneighbor graph, or random nearest-neighbor tree, whose geometry has been studied in several Euclidean settings [8, 9, 7, 4] . In the colored version, introduced by Preater and Aldous and further studied in [10, 1, 3], the focus is instead on the random partition generated by color inheritance. More precisely, in  \n∗ LPSM, Sorbonne Université, France, [anne.laure. basdevant@normalesup. org](anne.laure. basdevant@normalesup. org)  \n†CNRS and LMO, Université Paris-Saclay, France [arvind.singh@universite-paris-saclay.fr](arvind.singh@universite-paris-saclay.fr)  \nthe latter Euclidean model, the authors consider the unit cube [−1, 1] d. Initially, the origin is colored in blue and the boundary of the cube in red. Subsequently, independent, uniformly sampled random points fall in [−1, 1]d and, upon arrival, each point takes the color of the nearest point that has appeared so far. This procedure ultimately creates a random coloring of the unit cube 1 . One of the main results in [10, 3] is that there exists a.s. an open region containing the origin that is fully blue, i. e. , all red vertices remain at a positive distance from the initial blue vertex. On the other hand, the converse question of whether there may exist blue vertices arbitrarily close to the red boundary with positive probability is more delicate and still remains open.  \nIn contrast with these Euclidean works, the present paper deals with a discrete setting in which the underlying graph is a regular tree. This change of geometry leads to rather different phenomena and makes the genealogical structure of the process a natural object of study. Before specializing to trees, let us first describe the model on an arbitrary finite connected graph G = (V, E) . Given a set of initially colored vertices called seeds and a uniformly random permutation of the non-seed vertices (which encodes the order in which the vertices are revealed), the coloring procedure is defined as follows. When a vertex v is","cbCaiujJnWcdP4jB","https://ap.wps.com/l/cbCaiujJnWcdP4jB","pdf",1951029,1,39,"English","en",105,"# Introduction\n## Model on finite graphs\n## Arrival-time formulation and genealogy\n## Focus on regular trees","[{\"question\":\"How does the nearest-neighbor coloring process work?\",\"answer\":\"Vertices are revealed in a uniformly random order. When a vertex is revealed, it takes the color of one of its closest already colored vertices, chosen uniformly among ties.\"},{\"question\":\"What is special about the genealogy on an infinite regular tree?\",\"answer\":\"The genealogical graph is not connected. It contains infinitely many infinite one-ended components with distinct asymptotic directions, while each vertex has only finitely many descendants.\"},{\"question\":\"How are finitely many initial seeds handled and what is studied for finite trees?\",\"answer\":\"The structure of the genealogy is modified when starting from finitely many initial seeds. The paper also studies local limits as the height of finite regular trees grows, for two seed configurations: two fixed seeds, and one blue root seed with red leaf seeds.\"}]","Random Sequential Nearest-Neighbor Coloring on Trees | PDF",1787836001,98,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"random-sequential-nearest-neighbor-coloring-on-trees","",{"@graph":36,"@context":85},[37,54,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"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":53},"https://docshare.wps.com/document/random-sequential-nearest-neighbor-coloring-on-trees/151274/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":23,"description":14,"dateModified":62,"datePublished":62,"encodingFormat":61,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-27",true,{"@type":65,"interactionType":66,"userInteractionCount":4},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"How does the nearest-neighbor coloring process work?","Question",{"text":75,"@type":76},"Vertices are revealed in a uniformly random order. When a vertex is revealed, it takes the color of one of its closest already colored vertices, chosen uniformly among ties.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What is special about the genealogy on an infinite regular tree?",{"text":80,"@type":76},"The genealogical graph is not connected. It contains infinitely many infinite one-ended components with distinct asymptotic directions, while each vertex has only finitely many descendants.",{"name":82,"@type":73,"acceptedAnswer":83},"How are finitely many initial seeds handled and what is studied for finite trees?",{"text":84,"@type":76},"The structure of the genealogy is modified when starting from finitely many initial seeds. The paper also studies local limits as the height of finite regular trees grows, for two seed configurations: two fixed seeds, and one blue root seed with red leaf seeds.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"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":53,"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":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]