[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84485-en":3,"doc-seo-84485-105":29,"detail-sidebar-cat-0-en-105":82},{"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":13,"seo_description":14,"update_tm":27,"read_time":28},84485,549758146520,"Patrick","https://ap-avatar.wpscdn.com/avatar/80002397d8c0411e94?_k=1775819394049821470",8,"Research & Report","Dynamic Edge Coloring of Forests","Dynamic edge coloring maintains a proper κ-edge-coloring of a graph of maximum degree Δ under edge insertions and deletions, with κ=Δ+c and with recourse measured as the number of recolored edges per update. The work studies forests in both incremental and fully dynamic models. It analyzes greedy deterministic methods, gives tight recourse bounds, provides an optimal non-greedy algorithm on rooted fully dynamic forests, and develops a randomized distribution-maintaining algorithm with matching expected and lower-bound results.","arXiv :2605 .097 1 1v2 [ cs .DS] 11 Jul 2026  \nDynamic Edge Coloring of Forests  \nHaim Kaplan∗ David Naori∗ Yaniv Sadeh∗  \nAbstract  \nIn the dynamic edge coloring problem, one has to maintain a graph of maximum degree ∆ with at most ∆ + c colors, under edge updates. A prominent objective is to minimize therecourse, namely the number of edges that are recolored. We study this problem on forests, arguably the simplest graph class that already captures much of the complexity of the problem. We consider both the incremental model, where edges are only inserted and the fully dynamic model where edges may also be deleted.  \nIn the deterministic setting, we focus on the natural greedy algorithm. We show that it achieves O( c+∆ ) amortized recourse in the incremental model, and that this is tight up to tiebreaking. In contrast, in a fully dynamic forest, greedy can be forced to have Ω(log ∆ n) amortized recourse. To partially overcome this limitation of greedy within the deterministic setting, we give an optimal non-greedy algorithm with O(1) amortized recourse for rooted fully dynamic forests and c = ∆ − 2. In the randomized setting, we give a natural distribution-maintaining algorithm. In the incremental model, it achieves Θ( ) expected amortized recourse, and we show that this is optimal for every constant c. In the fully dynamic model, the same algorithm achieves Θ(min{~~∆~~c , log∆ n}) expected recourse for c > 0, and Θ(log∆ n) for c = 0 . We show that this is optimal for c = 0, and prove an Ω(1) lower bound for every constant c.  \n∗ The Blavatnik School of Computer Science and AI, Tel Aviv University, Tel Aviv, Israel. Emails: {haimk, dnaori, [yanivsadeh](yanivsadeh}@tau.ac.il. This)[}](yanivsadeh}@tau.ac.il. This)[@tau.ac.il](yanivsadeh}@tau.ac.il. This)[. This](yanivsadeh}@tau.ac.il. This) research was supported by ISF grant number 1156/23 and the Blavatnik research Foundation.  \nContents  \n1 Introduction 3  \n1.1 Related Work ........................................ 5  \n1.2 Preliminaries ........................................ 6  \n2 Technical Overview 7  \n2.1 Deterministic Algorithms ................................. 7  \n2.1.1 Incremental Forests (Insertions Only) ...................... 7  \n2.1.2 Fully Dynamic Forests (Insertions and Deletions) ................ 7  \n2.2 Randomized Algorithms .................................. 9  \n2.2.1 A Randomized Distribution-Maintaining Algorithm .............. 10  \n2.2.2 Recourse Analysis (Incremental and Fully Dynamic Forests) .......... 10  \n3 Deterministic Algorithms: A Formal Account 14  \n3.1 Incremental Forests (Insertions Only) .......................... 14  \n3.2 Fully Dynamic Forests (Insertions and Deletions) .................... 17  \n3.2.1 Greedy Logarithmic Amortized Recourse .................... 19  \n3.2.2 O(1) Recourse with 2∆ − 2 Colors on Rooted Forests .............. 24  \n3.3 Greedy Algorithms: Additional Variants and Discussion ................ 26  \n3.3.1 Greedy Optimality for ∆ = 2 ........................... 27  \n3.3.2 What Changes for the new Greedy Variants ................... 28  \n4 Randomized Algorithms: A Formal Account 30  \n4.1 A Randomized Distribution-Maintaining Algorithm ................... 30  \n4.1.1 Rooted Forests ................................... 30  \n4.1.2 General (unrooted) Forests via Rerooting .................... 35  \n4.2 Recourse Analysis ..................................... 36  \n4.2.1 Incremental Forests (Insertions Only) ...................... 37  \n4.2.2 Fully Dynamic Forests (Insertions and Deletions) ................ 42  \n5 Concluding Remarks 45  \nA Running Times 45  \nA.1 DistMaint Running Time ................................ 46  \nA.2 ColorfulPath Running Time .............................. 47  \nA.3 Greedy Family Running Times .............................. 48  \nB Additional Technical Analyses 49  \nB.1 Proving Lemma 3.3 ..................................... 49  \nB.2 Sublinear Worst-Case Recourse for ∆ Edge Coloring (used by Theorem 2.1) ..... 50","cbCaioO7MNctsvlV","https://ap.wps.com/l/cbCaioO7MNctsvlV","pdf",1001777,1,55,"English","en",105,"# Introduction\n# Technical Overview\n## Deterministic Algorithms\n## Randomized Algorithms\n# Deterministic Algorithms: A Formal Account\n## Incremental Forests (Insertions Only)\n## Fully Dynamic Forests (Insertions and Deletions)\n# Randomized Algorithms: A Formal Account\n## A Randomized Distribution-Maintaining Algorithm\n## Recourse Analysis\n# Concluding Remarks","[{\"question\":\"What recourse performance is achieved for greedy deterministic algorithms on forests?\",\"answer\":\"For incremental forests, the greedy algorithm achieves O(c+Δ) amortized recourse, which is essentially tight. In fully dynamic forests, greedy can be forced to incur Ω(log Δ · n) amortized recourse, motivating improved approaches.\"}]",1784195980,139,{"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":77,"head_meta":79,"extra_data":81,"updated_unix":27},"dynamic-edge-coloring-of-forests","",{"@graph":35,"@context":76},[36,53,67],{"@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/dynamic-edge-coloring-of-forests/84485/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":61,"encodingFormat":60,"isAccessibleForFree":62,"interactionStatistic":63},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-16",true,{"@type":64,"interactionType":65,"userInteractionCount":4},"InteractionCounter",{"@type":66},"ViewAction",{"@type":68,"mainEntity":69},"FAQPage",[70],{"name":71,"@type":72,"acceptedAnswer":73},"What recourse performance is achieved for greedy deterministic algorithms on forests?","Question",{"text":74,"@type":75},"For incremental forests, the greedy algorithm achieves O(c+Δ) amortized recourse, which is essentially tight. In fully dynamic forests, greedy can be forced to incur Ω(log Δ · n) amortized recourse, motivating improved approaches.","Answer","https://schema.org",{"og:url":51,"og:type":78,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":80,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":83},[84,88,92,96,101,106,111,114,119,122,126],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":85,"show_sort_weight":86,"slug":87},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":89,"show_sort_weight":90,"slug":91},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":93,"show_sort_weight":94,"slug":95},"Exam",70,"exam",{"id":97,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},5,"Comic",60,"comic",{"id":102,"doc_module":4,"doc_module_name":45,"category_name":103,"show_sort_weight":104,"slug":105},6,"Technology",50,"technology",{"id":107,"doc_module":4,"doc_module_name":45,"category_name":108,"show_sort_weight":109,"slug":110},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":112,"slug":113},30,"research-report",{"id":115,"doc_module":4,"doc_module_name":45,"category_name":116,"show_sort_weight":117,"slug":118},9,"Religion & Spirituality",20,"religion-spirituality",{"id":117,"doc_module":4,"doc_module_name":45,"category_name":120,"show_sort_weight":117,"slug":121},"World Cup","world-cup",{"id":123,"doc_module":4,"doc_module_name":45,"category_name":124,"show_sort_weight":123,"slug":125},10,"Lifestyle","lifestyle",{"id":127,"doc_module":4,"doc_module_name":45,"category_name":128,"show_sort_weight":97,"slug":129},19,"General","general"]