[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-150545-en":3,"doc-seo-150545-105":30,"detail-sidebar-cat-0-en-105":92},{"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":27,"seo_description":14,"update_tm":28,"read_time":29},150545,7971461741311,"Ophelia","https://ap-avatar.wpscdn.com/avatar/74000253aff267980c6?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779345379180704826",8,"Research & Report","Weak Coloring Numbers of Minor-Closed Graph Classes - Main Results","We analyze how weak coloring numbers grow in graph classes that exclude a fixed graph as a minor. Building on results of Van den Heuvel et al. on polynomial growth in r, the paper refines the polynomial bound up to an O(r log r) factor and links the exponent to the minor’s structural parameter, 2-treedepth. For X-minor-free graphs G, it proves wcolr(G) = O(rtd(X)−1 log r), improving prior exponential-type bounds. For planar graphs with bounded treewidth, the maximum r-th weak coloring number is O(r2 log r), matching known lower bounds.","arXiv :2407 .04588v2 [math .CO] 4 Apr 2025  \nWEAK COLORING NUMBERS OF MINOR-CLOSED GRAPH CLASSES  \nJĘDRZEJ HODOR, HOANG LA, PIOTR MICEK, AND CLÉMENT RAMBAUD  \nAbstract . We study the growth rate of weak coloring numbers of graphs excluding a fixed graph as a minor. Van den Heuvel et al. (European J. of Combinatorics, 2017) showed that for a fixed graph X , the maximum r-th weak coloring number of X-minor-free graphs is polynomial in r. We determine this polynomial up to a factor of O (r log r) . Moreover, we tie the exponent of the polynomial to a structural property of X , namely, 2-treedepth. As a result, for a fixed graph X and an X-minor-free graph G, we show that wcolr (G) = O(rtd(X)−1 log r), which improves on the bound wcolr (G) = O(rg(td(X))) given by Dujmović et al. (SODA, 2024), where g is an exponential function. In the case of planar graphs of bounded treewidth, we show that the maximum r-th weak coloring number is in O (r2 log r), which is best possible.  \n(J. Hodor) Theoretical Computer Science Department, Faculty of Mathematics and Computer Science and Doctoral School of Exact and Natural Sciences, Jagiellonian University, Kraków, Poland  \n(H. La) LISN, Université Paris-Saclay, CNRS, Gif-sur-Yvette, France  \n(P. Micek) Theoretical Computer Science Department, Faculty of Mathematics and Computer Science, Jagiellonian University, Kraków, Poland  \n(C. Rambaud) Université Côte d’Azur, CNRS, Inria, I3S, Sophia-Antipolis, France E-mail addresses: [jedrzej.hodor@gmail.com](jedrzej.hodor@gmail.com) , [hoang.la.research@gmail.com](hoang.la.research@gmail.com) ,  \n[piotr.micek@uj.edu.pl](piotr.micek@uj.edu.pl) , [clement.rambaud@inria.fr](clement.rambaud@inria.fr) .  \nThis research was funded by the National Science Center of Poland under grant UMO-2023/05/Y/ST6/00079 within the WEAVE-UNISONO program. J. Hodor was partially supported by a Polish Ministry of Education and Science grant (Perły Nauki; PN/01/0265/2022) . C. Rambaud was partially supported by the French Agence Nationale de la Recherche under contract Digraphs ANR-19-CE48-0013-01 .  \n2 HODOR, LA, MICEK, AND RAMBAUD  \n1. Introduction  \nLet G be a graph, let Π(G) be the set of all vertex orderings of G, let σ ∈ Π(G), and let r bea nonnegative integer. For all u and v vertices of G, we say that v is weakly r-reachable from u in (G,σ), if there exists a path between u and v in G containing at most r edges such that for every vertex w on the path, v ⩽σ w. Let WReachr[G,σ, u] be the set of vertices that are weakly r-reachable from u in (G,σ) . The r-th weak coloring number of G is defined as  \nwcolr(G) = min max |WReachr[G,σ, u]| .  \nσ∈Π(G) u∈V(G)  \nLet X be a graph. The treedepth of X , denoted by td(X), is defined recursively as follows  \n􀀸􀀾  \ntd(X) = 􀀼􀀾􀀺  \n0 if X is the null graph, min v∈V(X) td(X − v) + 1 if X is connected 1 , and  \nmaxi∈[k] td(Ci) if X consists of components C1 ,..., Ck and k > 1.  \nThe following two theorems are among the main contributions of this paper.  \nTheorem 1 . For every positive integer t, for every graph X with td(X) ⩽ t, there exists an integer c such that for every graph G, if G is X-minor-free, then for every integer r with r ⩾ 2 ,  \nwcolr (G) ⩽ c · rt−1 log r.  \nTheorem 2 . For every integer t with t ⩾ 2, for every graph X with td(X) ⩽ t, there existsan integer c such that for every graph G, if G is X-minor-free, then for every integer r with r ⩾ 2 ,  \nwcolr(G) ⩽ c · (tw(G) + 1) · rt−2 log r.  \nWeak coloring numbers were introduced by Kierstead and Yang [14] in 2003, though a parameter similar to wcol2 (G) is already present in the work of Chen and Schelp [1] from 1993 . This family of parameters gained considerable attention when Zhu [21] proved that it captures important and robust notions of sparsity, namely, bounded expansion and nowhere denseness. Specifically, a class of graphs C has bounded expansion if and only if there exists a function g such that for every graph G in C and every positive integer r, we have wcolr(G) ⩽ g(","cbCaich72Gi562c8","https://ap.wps.com/l/cbCaich72Gi562c8","pdf",1528636,1,50,"English","en",105,"# Introduction\n## Weak coloring numbers and reachability\n## Treedepth and structural parameters\n## Main theorems and improvements\n## Related work and applications","[{\"question\":\"What are weak coloring numbers and how are they defined?\",\"answer\":\"The r-th weak coloring number wcol_r(G) is defined via the minimum, over all vertex orderings σ, of the maximum size of the set of vertices that are weakly r-reachable from each vertex u using paths with at most r edges while respecting the ordering constraint.\"},{\"question\":\"What is the paper’s main improvement for X-minor-free graphs?\",\"answer\":\"For a fixed graph X and an X-minor-free graph G, the paper shows wcol_r(G) = O(rtd(X)−1 log r). This improves earlier bounds that involved an exponential function g tied to the treedepth-related parameter.\"},{\"question\":\"What bound is obtained for planar graphs of bounded treewidth?\",\"answer\":\"For planar graphs with bounded treewidth, the paper proves the maximum r-th weak coloring number is in O(r2 log r), and states this rate is best possible.\"}]","Weak Coloring Numbers of Minor-Closed Graph Classes - Main Results | PDF",1787822438,126,{"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":87,"head_meta":89,"extra_data":91,"updated_unix":28},"weak-coloring-numbers-of-minor-closed-graph-classes-main-results","",{"@graph":36,"@context":86},[37,54,69],{"@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/weak-coloring-numbers-of-minor-closed-graph-classes-main-results/150545/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":23,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-09-04","2026-08-27",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"What are weak coloring numbers and how are they defined?","Question",{"text":76,"@type":77},"The r-th weak coloring number wcol_r(G) is defined via the minimum, over all vertex orderings σ, of the maximum size of the set of vertices that are weakly r-reachable from each vertex u using paths with at most r edges while respecting the ordering constraint.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"What is the paper’s main improvement for X-minor-free graphs?",{"text":81,"@type":77},"For a fixed graph X and an X-minor-free graph G, the paper shows wcol_r(G) = O(rtd(X)−1 log r). This improves earlier bounds that involved an exponential function g tied to the treedepth-related parameter.",{"name":83,"@type":74,"acceptedAnswer":84},"What bound is obtained for planar graphs of bounded treewidth?",{"text":85,"@type":77},"For planar graphs with bounded treewidth, the paper proves the maximum r-th weak coloring number is in O(r2 log r), and states this rate is best possible.","https://schema.org",{"og:url":52,"og:type":88,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":90,"canonical":52},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":93},[94,98,102,106,111,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":46,"category_name":95,"show_sort_weight":96,"slug":97},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":99,"show_sort_weight":100,"slug":101},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":103,"show_sort_weight":104,"slug":105},"Exam",70,"exam",{"id":107,"doc_module":4,"doc_module_name":46,"category_name":108,"show_sort_weight":109,"slug":110},5,"Comic",60,"comic",{"id":112,"doc_module":4,"doc_module_name":46,"category_name":113,"show_sort_weight":21,"slug":114},6,"Technology","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":107,"slug":138},19,"General","general"]