[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-135383-en":3,"doc-seo-135383-105":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},135383,962075114765,"Quinn","https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd",8,"Research & Report","Mycielski graphs and PR proofs","Mycielski graphs are triangle-free graphs Mk with arbitrarily large chromatic number, and although their k-chromaticity has an informal argument, automated proof search has difficulty establishing it. The paper analyzes the complexity of clausal proofs for the uncolorability of Mk using k−1 colors, focusing on variants of the PR (propagation redundancy) system without new variables and optionally without deletion. It presents a sublinear-length, constant-width PR proof with neither new variables nor deletion, implements and verifies a proof generator, and studies CDCL performance on clause-extended formulas.","Mycielski graphs and PR proofs  \nEmre Yolcu, Xinyu Wu, and Marijn J. H. Heule  \nCarnegie Mellon University, Pittsburgh, PA 15213, USA {emreyolcu,xinyuwu,[marijn}@cmu.edu](marijn}@cmu.edu)  \nAbstract. Mycielski graphs are a family of triangle-free graphs Mk with arbitrarily high chromatic number. Mk has chromatic number k and there is a short informal proof of this fact, yet finding proofs of it via automated reasoning techniques has proved to be a challenging task. In this paper, we study the complexity of clausal proofs of the uncolorability of Mk with k 􀀀 1 colors. In particular, we consider variants of the PR (propagation redundancy) proof system that are without new variables, and with or without deletion. These proof systems are of interest due to their potential uses for proof search. As our main result, we present a sublinear-length and constant-width PR proof without new variables or deletion. We also implement a proof generator and verify the correctness of our proof. Furthermore, we consider formulas extended with clauses from the proof until a short resolution proof exists, and investigate the performance of CDCL in finding the short proof. This turns out to be difficult for CDCL with the standard heuristics. Finally, we describe an approach inspired by SAT sweeping to find proofs of these extended formulas.  \n1 Introduction  \nProof complexity investigates the relative strengths of Cook–Reckhow proof systems [7], defined in terms of the length of the shortest proof of a tautology as a function of the length of the tautology. Proof systems are separated with respect to their strengths by establishing lower and upper bounds on the lengths of the proofs of certain “difficult” tautologies in each system. Finding short proofs of such tautologies in a proof system is a method for proving small upper bounds, which provide evidence for the strength of a proof system. Similarly, the existence of a large lower bound implies that a proof system is relatively weak. The related field of SAT solving involves the study of search algorithms that have corresponding proof systems, and concerns itself with not only the existence of short proofs, but also the prospect of finding them automatically when they exist. As a result, the two areas interact. The long-term agenda of proof complexity is to prove lower bounds on proof systems of increasing strength towards concluding NP  co-NP, whereas SAT solving benefits from strong proof systems with properties that make them suitable for automation. A recently proposed such system is PR (propagation redundancy) [14] and some of its variants SPR (subset PR), PR 􀀀(without new variables), DPR 􀀀 (allowing deletion) .  \n2 E. Yolcu et al.  \nFor several difficult tautologies, PR has been shown to admit proofs that are short (at most polynomial-length), narrow (small clause width), and without extension (disallowing new variables) [5, 12 , 13 , 14] . From the perspective of proof search, these are favorable qualities for a proof system:  \n– Polynomial-length is essentially a necessity.  \n– Small width implies that we may limit the search to narrow proofs.  \n– Eliminating extension drastically shrinks the search space.  \nCompared to strong proof systems with extension, a proof system with the above properties may admit a proof search algorithm that is effective in practice.  \nMycielski graphs are a family of triangle-free graphs Mk with arbitrarily high chromatic number. In particular, Mk has chromatic number k. Despite having a simple informal proof, this has been a difficult fact to prove via automated reasoning techniques, and the state-of-the-art tools can only handle instances up to M6 or M7 [6, 9 , 18 , 19 , 20 , 21 , 23] . Symmetry breaking [8], a crucial automated reasoning technique for hard graph coloring instances, is hardly effective on these graphs as the largest clique has size 2 . Most short PR proofs for hard problems are based on symmetry arguments. Donald Knuth challenged us in personal communication","cbCaiqTZAnc5YqeK","https://ap.wps.com/l/cbCaiqTZAnc5YqeK","pdf",499217,1,17,"English","en",105,"# Introduction\n# Preliminaries","[{\"question\":\"What problem does the paper study regarding Mycielski graphs?\",\"answer\":\"It studies the complexity of clausal proofs showing that Mycielski graphs Mk are not colorable with k−1 colors, within specific proof systems.\"},{\"question\":\"What variants of PR proof systems are investigated?\",\"answer\":\"The paper considers PR variants that disallow new variables and also examines versions that allow or disallow deletion, aiming to understand their proof lengths and widths.\"},{\"question\":\"What is the main proof result and how is it validated?\",\"answer\":\"The authors provide a sublinear-length and constant-width PR proof without new variables or deletion, then implement a proof generator and verify the generated proofs for correctness.\"}]","Mycielski graphs and PR proofs | PDF",1787310477,43,{"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},"mycielski-graphs-and-pr-proofs","",{"@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/mycielski-graphs-and-pr-proofs/135383/",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","2026-08-23","2026-08-21",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 problem does the paper study regarding Mycielski graphs?","Question",{"text":76,"@type":77},"It studies the complexity of clausal proofs showing that Mycielski graphs Mk are not colorable with k−1 colors, within specific proof systems.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"What variants of PR proof systems are investigated?",{"text":81,"@type":77},"The paper considers PR variants that disallow new variables and also examines versions that allow or disallow deletion, aiming to understand their proof lengths and widths.",{"name":83,"@type":74,"acceptedAnswer":84},"What is the main proof result and how is it validated?",{"text":85,"@type":77},"The authors provide a sublinear-length and constant-width PR proof without new variables or deletion, then implement a proof generator and verify the generated proofs for correctness.","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},[94,98,102,106,111,116,121,124,129,132,136],{"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":114,"slug":115},6,"Technology",50,"technology",{"id":117,"doc_module":4,"doc_module_name":46,"category_name":118,"show_sort_weight":119,"slug":120},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":122,"slug":123},30,"research-report",{"id":125,"doc_module":4,"doc_module_name":46,"category_name":126,"show_sort_weight":127,"slug":128},9,"Religion & Spirituality",20,"religion-spirituality",{"id":127,"doc_module":4,"doc_module_name":46,"category_name":130,"show_sort_weight":127,"slug":131},"World Cup","world-cup",{"id":133,"doc_module":4,"doc_module_name":46,"category_name":134,"show_sort_weight":133,"slug":135},10,"Lifestyle","lifestyle",{"id":137,"doc_module":4,"doc_module_name":46,"category_name":138,"show_sort_weight":107,"slug":139},19,"General","general"]