Hypergraph Defect Tolerance for Multi-Center Bonding Materials: Tight Bounds, Optimal Constructions, and Sharp Phase Transitions

Hypergraph Defect Tolerance for Multi-Center Bonding Materials: Tight Bounds, Optimal Constructions, and Sharp Phase Transitions | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Hypergraph Defect Tolerance for Multi-Center Bonding Materials: Tight Bounds, Optimal Constructions, and Sharp Phase Transitions Satish Prajapati This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9600963/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Defect tolerance in physical systems has been extensively studied using bipartite graph models, where each functional unit comprises exactly two components. This pairwise assumption fundamentally fails for materials with multi-center bonding, where functional units involve three or more atomic sites. Examples include three-center two electron bonds in boranes [29], four-center bonds in hypervalent compounds [44], and six-center octahedral coordination in SF6 [45]. We present a rigorous generalization using k-uniform hypergraphs and establish a tight lower bound m(k,t,q) ≥ k k−1· t q−1 for the minimum redundancy m required to tolerate t defects in a system with q component types. This bound proves a (k − 1)-fold improvement over naively applying pairwise models and establishes the first rigorous connection between defect tolerance and the strong chromatic number ¯χ(H) [8, 9]. We construct optimal hypergraphs achieving this bound for (k,q) = (3,2) using the Fano plane [15], (4,2) using PG(2,3) [16], and (3,3) using AG(2,3) [17]. Probabilistic methods [10, 12] demonstrate asymptotic existence for all parameters, and we characterize a sharp phase transition at mc ≈ 10 for (k = 3, q = 2). This framework provides a mathematical foundation for designing defect tolerant boranes [? ], perovskites [34, 35, 36], octahedral coordination compounds [46], high-entropy alloys [49, 50, 51], metallic glasses [52, 53, 54], MAX phases [55, 56], quantum materials [57, 58], and two-dimensional materials [62, 63, 64]. Materials Theory and Modeling Materials Engineering Hypergraph defect tolerance multi-center bonding strong chromatic number Fano plane finite projective geometry Steiner triple system combinatorial design redundancy optimization phase transition probabilistic method Lovász Local Lemma Erdős–Ko–Rado theorem boranes three-center two-electron bonds perovskites ABX₃ structure octahedral coordination sulfur hexafluoride hypervalent compounds high-entropy alloys metallic glasses MAX phases quantum materials topological insulators two-dimensional materials graphene transition metal dichalcogenides defect physics fault-tolerant materials computational materials science combinatorial materials discovery finite geometries affine plane projective plane information-theoretic bounds scaling laws redundancy savings tight lower bound optimal construction random hypergraph covering design intersecting family t-defect tolerance component types primary nodes redundant nodes hyperedge functional unit bond order coordination numbe Full Text Additional Declarations The authors declare no competing interests. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-9600963","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":633651680,"identity":"08f2c5b1-9e06-4af5-b1d3-5b07574f7941","order_by":0,"name":"Satish Prajapati","email":"data:image/png;base64,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","orcid":"https://orcid.org/0009-0006-3801-1137","institution":"Government College of Engineering and Ceramic Technology, Kolkata, India","correspondingAuthor":true,"prefix":"","firstName":"Satish","middleName":"","lastName":"Prajapati","suffix":""}],"badges":[],"createdAt":"2026-05-03 15:17:07","currentVersionCode":1,"declarations":{"humanSubjects":false,"vertebrateSubjects":true,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":true},"doi":"10.21203/rs.3.rs-9600963/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9600963/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":108804785,"identity":"72313dd6-002c-4182-99b8-84706b7780a1","added_by":"auto","created_at":"2026-05-08 15:23:32","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":17412599,"visible":true,"origin":"","legend":"","description":"","filename":"mainmanuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9600963/v1_covered_4527b943-a919-43a1-b463-25a2d475d1db.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003eHypergraph Defect Tolerance for Multi-Center Bonding Materials: Tight Bounds, Optimal Constructions, and Sharp Phase Transitions\u003c/p\u003e","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"Government College of Engineering and Ceramic Technology","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Hypergraph defect tolerance, multi-center bonding, strong chromatic number, Fano plane, finite projective geometry, Steiner triple system, combinatorial design, redundancy optimization, phase transition, probabilistic method, Lovász Local Lemma, Erdős–Ko–Rado theorem, boranes, three-center two-electron bonds, perovskites, ABX₃ structure, octahedral coordination, sulfur hexafluoride, hypervalent compounds, high-entropy alloys, metallic glasses, MAX phases, quantum materials, topological insulators, two-dimensional materials, graphene, transition metal dichalcogenides, defect physics, fault-tolerant materials, computational materials science, combinatorial materials discovery, finite geometries, affine plane, projective plane, information-theoretic bounds, scaling laws, redundancy savings, tight lower bound, optimal construction, random hypergraph, covering design, intersecting family, t-defect tolerance, component types, primary nodes, redundant nodes, hyperedge, functional unit, bond order, coordination numbe","lastPublishedDoi":"10.21203/rs.3.rs-9600963/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9600963/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eDefect tolerance in physical systems has been extensively studied using bipartite graph models, where each functional unit comprises exactly two components. 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