[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81646-en":3,"doc-seo-81646-105":29,"detail-sidebar-cat-0-en-105":90},{"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},81646,16904993612988,"Olivia Brown","https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd",8,"Research & Report","Sampling Simultaneous Edge Colorings","Study of the sampling problem for simultaneous edge colorings for two graphs sharing a vertex set. A simultaneous coloring assigns colors to edges of the union so that each original graph is properly colored. For graphs with maximum degree Δ, results analyze Markov chains that sample uniformly, proving rapid mixing of Glauber and flip dynamics via coupling using a weighted Hamming distance. Mixing time becomes O(mlog n) in regimes determined by Δ rather than 4Δ, with explicit k thresholds.","arXiv :2605 .05046v2 [ cs .DM] 10 Jul 2026  \nSampling Simultaneous Edge-Colorings  \nEzra Furtado-Tiwari and Eric Vigoda ∗  \nAbstract  \nWe study the sampling problem for simultaneous edge colorings. Given a pair of graphs G 1 = (V, E1 ) and G2 = (V, E2 ) which are on the same vertex set V , a simultaneous edge coloring is an edge coloring of G 1 ∪G2 so that each of the individual graphs is properly colored. When each of G 1 and G2 are of maximum degree ∆, then it is conjectured that ∆ + 2 colors suffice, and recent work asymptotically establishes the conjecture.  \nWe study Markov chains for randomly sampling from the uniform distribution over simultaneous edge colorings. Straightforward applications of Jerrum’s classical coupling argument establish rapid mixing of the Glauber dynamics on the corresponding line graph when k > 8∆ . We present a simple weighted Hamming distance for which Jerrum’s coupling yields optimal mixing time (up to constant factors) of O (mlog n) when k > (6 + δ)∆ for any fixed δ > 0. Moreover, utilizing the flip dynamics with our new metric, we obtain O(mlog n) mixing of the flip dynamics when k ≥ 5.948∆, using a local choice of flip parameters which only flips bounded-size components. The proof adapts previous coupling analyses for the flip dynamics to the setting of simultaneous edge colorings.  \n∗ Department of Computer Science, University of California, Santa Barbara. Email: {ezrafurtadotiwari,[vigoda](vigoda}@ucsb.edu. Research)[}](vigoda}@ucsb.edu. Research)[@ucsb.edu. Research](vigoda}@ucsb.edu. Research) supported in part by NSF grant CCF-2147094.  \n1 Introduction  \nA natural combinatorial problem which was recently introduced with intriguing open problems is simultaneous edge colorings. In the simultaneous edge coloring problem, we are given as input a pair of undirected graphs G1 = (V, E1 ) and G2 = (V, E2 ) on a common vertex set V = [n] = {1,..., n}, where each graph has maximum degree of ∆ . A simultaneous edgecoloring is an assignment ϕ : E1 ∪ E2 → [k] such that for all i ∈ {1 , 2}, all e, e′ ∈ Ei , if e ∩ e′  ∅ then ϕ(e)  ϕ (e′) . In other words we are coloring the edges of both graphs so that the edges within each graph are properly colored, but adjacent edges in different graphs can be monochromatic.  \nLet χ (G1 , G2 ) denote the minimum k for which a simultaneous k-edge-coloring of G 1 and G2 exists. Vizing’s Theorem shows that for a graph G of maximum degree ∆, its edge chromatic number satisfies χ (G) ≤ ∆ + 1 . The intriguing question for simultaneous edge colorings is whether χ(G1 , G2 ) ∼ ∆ .  \nThe simultaneous edge coloring problem for graphs G 1 = (V, E1 ) and G2 = (V, E2 ) can be recast as a vertex coloring problem via line graphs. Let L 1 and L2 denote the line graphs ofofE2GL11AaasnniddmG2L2ultreanesnopduesctideiedvelntgeiyfyc,ioandngloricoverngntosicfideesG1r thecorreandgrspG2aopnishdti(di,)seobthnttaattoinedappa prbeoyapretrakinveingbotrtethe uh E1x coloniarionnndgofm.  d2,G2anhdave mhenceaximum deG has magxree ∆,imumtdhenegreeeacathomfosLt14L42 has maxHoweveri,mum degrethis reducetiatonobG2unsl,ceuranssesdthieempdgey cosorrtararentisspisnotgnructufromd toedhe:iaffsdjacerenameetncgroriyagarimpedgises oosee TnnhlyousaroddwmithiilcoeonsnalthtrecaoninnsaitstrvewairithntseduinoctGnio1oreachn suwoggiththesietnrsa dependence on 4∆, the underlying constraint system retains a more refined structure tied to the original degree ∆ .  \nThis observation raises the question of whether the mixing behavior of associated Markov chains for randomly sampling simultaneous edge colorings is governed by the maximum degree 4∆ of the union graph, or by the intrinsic constraint degree ∆ of the original graphs. Our results show that the latter is the case: optimal mixing time can be achieved in a regime determined by ∆, rather than 4∆ .  \nThe problem of simultaneous edge coloring was introduced recently by Cabello (see Bousquet and Durain [BD20]) who conjectured that χ (G1 , G2 ) ≤ ∆ + 2 for any graphs ","cbCairQzgL5YZkiv","https://ap.wps.com/l/cbCairQzgL5YZkiv","pdf",361421,1,25,"English","en",105,"# Abstract\n# Introduction\n## Problem definition and motivation\n## Markov chain sampling approach","[{\"question\":\"What is a simultaneous edge coloring for two graphs on the same vertex set?\",\"answer\":\"It is an assignment of colors to edges of the union graph such that, within each input graph, adjacent edges receive different colors. Edges from different graphs may be monochromatic.\"},{\"question\":\"What conjecture and threshold are discussed for the number of colors k?\",\"answer\":\"It is conjectured that for maximum degree Δ, Δ+2 colors should suffice for simultaneous edge colorings. Prior work establishes the conjecture asymptotically, and the paper investigates sampling efficiency in related k vs. Δ regimes.\"},{\"question\":\"Which Markov chains are analyzed for sampling uniformly from simultaneous edge colorings?\",\"answer\":\"The paper studies the Glauber dynamics and a flip dynamics. Using coupling and a weighted Hamming distance, it derives rapid mixing time bounds such as O(mlog n) when k exceeds explicit multiples of Δ.\"}]",1784175129,63,{"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":85,"head_meta":87,"extra_data":89,"updated_unix":27},"sampling-simultaneous-edge-colorings","",{"@graph":35,"@context":84},[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/sampling-simultaneous-edge-colorings/81646/",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",[70,76,80],{"name":71,"@type":72,"acceptedAnswer":73},"What is a simultaneous edge coloring for two graphs on the same vertex set?","Question",{"text":74,"@type":75},"It is an assignment of colors to edges of the union graph such that, within each input graph, adjacent edges receive different colors. Edges from different graphs may be monochromatic.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"What conjecture and threshold are discussed for the number of colors k?",{"text":79,"@type":75},"It is conjectured that for maximum degree Δ, Δ+2 colors should suffice for simultaneous edge colorings. Prior work establishes the conjecture asymptotically, and the paper investigates sampling efficiency in related k vs. Δ regimes.",{"name":81,"@type":72,"acceptedAnswer":82},"Which Markov chains are analyzed for sampling uniformly from simultaneous edge colorings?",{"text":83,"@type":75},"The paper studies the Glauber dynamics and a flip dynamics. Using coupling and a weighted Hamming distance, it derives rapid mixing time bounds such as O(mlog n) when k exceeds explicit multiples of Δ.","https://schema.org",{"og:url":51,"og:type":86,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":88,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":91},[92,96,100,104,109,114,119,122,127,130,134],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":93,"show_sort_weight":94,"slug":95},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":97,"show_sort_weight":98,"slug":99},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":101,"show_sort_weight":102,"slug":103},"Exam",70,"exam",{"id":105,"doc_module":4,"doc_module_name":45,"category_name":106,"show_sort_weight":107,"slug":108},5,"Comic",60,"comic",{"id":110,"doc_module":4,"doc_module_name":45,"category_name":111,"show_sort_weight":112,"slug":113},6,"Technology",50,"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":125,"slug":126},9,"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":45,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":45,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":45,"category_name":136,"show_sort_weight":105,"slug":137},19,"General","general"]