[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85006-en":3,"doc-seo-85006-105":29,"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":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},85006,13056703019404,"Miles","https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0",8,"Research & Report","Unconditional Lower Bounds for Degree Fault Tolerant Spanners","Study multiplicative graph spanners in the f-degree fault tolerant (f-DFT) model, requiring distance approximation after any temporary failure of an edge subset whose maximum degree is f. Establish n-vertex graphs forcing every f-DFT (2k−1)-stretch spanner H to have size at least Ω(f^{1−1/kn} n^{1+1/k}). The argument removes the girth-conjecture dependency by giving an unconditional version that also matches known upper bounds up to exp(k).","arXiv :2607 .07576v 1 [ cs .DS] 8 Jul 2026  \nUnconditional Lower Bounds for Degree Fault Tolerant  \nSpanners  \nGreg Bodwin and Aleksey Lopez  \nUniversity of Michigan EECS  \n{bodwin, [lopezag](lopezag}@umich. edu)[}](lopezag}@umich. edu)[@umich. edu](lopezag}@umich. edu)  \nAbstract  \nWe study multiplicative graph spanners in the f-degree fault tolerant (f-DFT) model, in which the spanner must approximately preserve distances even after any subset of edges of maximum degree f temporarily “fails” and is removed from the graph. We prove that there are n-node lower bound graphs for which any f-DFT (2k − 1)-stretch spanner H must have size  \n|E(H)| ≥ Ω 􀀐 f 1−1/kn1+1/k􀀑 .  \nThis matches a lower bound that was previously only known to hold conditionally, under the 1963 girth conjecture of Erd˝os. It also matches the current upper bounds, up to a factor of exp (k) . Our proof is an analysis of the so-called Wenger graphs (J. Comb. Theory 1991), via their recent reinterpretation by Szab´o and by Conlon (Am. Math. Monthly 2021) .  \n1 Introduction  \n1.1 Spanners and the Girth Conjecture  \nA multiplicative spanner is a primitive in graph theory and algorithms, which sparsifies an input graph while approximately preserving its shortest path distances. Spanners were abstracted in the late ’80s after being implicitly studied in networking, distributed algorithms, etc (see survey [1]) .  \nDefinition 1 (Spanners [29]) . Given a graph G = (V, E, w), a subgraph H = (V, E′, w) is ak-spanner of G if for all nodes s, t, we have distH (s, t) ≤ k · distG (s, t) .  \nSpanners and their variants are often studied from the extremal perspective, where the goal is to prove that one can always construct a spanner with a good tradeoff between size and error. The following classic theorem of Alth¨ofer, Das, Dobkin, Joseph, and Soares [2] conditionally settles the extremal tradeoff available for multiplicative spanners:  \nTheorem 2 ([2]) . For all positive integers n, k, every n-node graph has a (2k − 1) -spanner on O (n1+1/k ) edges. Moreover, assuming the girth conjecture [18], for all positive integers k there are families of n-node graphs on which this size bound cannot be improved to o (n1+1/k ) .  \nThe girth conjecture referenced in this theorem, attributed to Erd˝os [18], says the following. Let us denote by ex(n, C≤2k) the maximum possible number of edges in an n-node graph that does not have any cycles of length ≤ 2k as subgraphs. There is a simple folklore counting argument, called the Moore bound, that implies  \nex (n, C≤2k) ≤ O 􀀐n 1+1/k􀀑 .  \nThe girth conjecture posits that this bound is asymptotically best possible for all k. The girth conjecture is an old and very difficult problem in extremal graph theory, likely beyond the reach of current techniques. It has been confirmed only for k ∈ {1, 2 , 3 , 5} [34, 35], with no new cases settled since the 1950s. In fact, the community has more recently started to regard it with some skepticism. For example, in an article from 2021, Conlon [14] writes:  \n“It is now quite commonplace to believe that the true value of certain extremal numbers lie below what the classical arguments give. We suspect that this should already be the case for C8 , that is,  \nthat ex(n, C8 ) = o(n5/4) .”  \nHere ex(n, C8 ) is the maximum possible number of edges in an n-node graph with no C8 subgraph. It follows from the definitions that ex (n, C8 ) ≥ ex(n, C≤8), and so the suggested bound of o (n5/4) would imply that the girth conjecture is false already for k = 4 . Since this seems difficult to prove, Conlon goes on to give a more attainable conjecture, very roughly stating that the techniques from incidence geometry used to confirm the girth conjecture for k ∈ {1, 2 , 3 , 5} cannot be extended to other values of k. This conjecture is related to the classical Feit-Higman Theorem [19], which is sometimes interpreted as further evidence against the girth conjecture.1 For further technical discussion and references on the girth conj","cbCaidbJNJ9cOLfL","https://ap.wps.com/l/cbCaidbJNJ9cOLfL","pdf",474997,1,14,"English","en",105,"# Abstract\n# Introduction\n## Spanners and the Girth Conjecture\n## Fault Tolerant Spanners","[{\"question\":\"What does the f-degree fault tolerant (f-DFT) spanner model require?\",\"answer\":\"It requires a spanner to preserve approximate shortest-path distances even after any subset of edges of maximum degree f is removed due to temporary failure.\"},{\"question\":\"What lower bound is proved for f-DFT (2k−1)-stretch spanners?\",\"answer\":\"For certain n-node graphs, any such spanner H must have size |E(H)| ≥ Ω(f^{1−1/kn} n^{1+1/k}).\"},{\"question\":\"How does the paper relate its proof to Wenger graphs and the girth conjecture?\",\"answer\":\"The result matches bounds previously known only conditionally under Erdős’s 1963 girth conjecture, and the proof analyzes Wenger graphs using a recent reinterpretation by Szabó and Conlon.\"}]",1784200193,35,{"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":86,"head_meta":88,"extra_data":90,"updated_unix":27},"unconditional-lower-bounds-for-degree-fault-tolerant-spanners","",{"@graph":35,"@context":85},[36,53,68],{"@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/unconditional-lower-bounds-for-degree-fault-tolerant-spanners/85006/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","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 the f-degree fault tolerant (f-DFT) spanner model require?","Question",{"text":75,"@type":76},"It requires a spanner to preserve approximate shortest-path distances even after any subset of edges of maximum degree f is removed due to temporary failure.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What lower bound is proved for f-DFT (2k−1)-stretch spanners?",{"text":80,"@type":76},"For certain n-node graphs, any such spanner H must have size |E(H)| ≥ Ω(f^{1−1/kn} n^{1+1/k}).",{"name":82,"@type":73,"acceptedAnswer":83},"How does the paper relate its proof to Wenger graphs and the girth conjecture?",{"text":84,"@type":76},"The result matches bounds previously known only conditionally under Erdős’s 1963 girth conjecture, and the proof analyzes Wenger graphs using a recent reinterpretation by Szabó and Conlon.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"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":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":45,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":45,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":45,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":45,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":45,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]