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Hence, (ii) each partial sum" /> <meta name="twitter:image" content="http://a.academia-assets.com/images/twitter-card.jpeg" /> <meta property="fb:app_id" content="2369844204" /> <meta property="og:type" content="article" /> <meta property="og:url" content="https://www.academia.edu/65089431/Robust_cycle_bases_do_not_exist_for_K_n_n_if_n_8" /> <meta property="og:title" content="Robust cycle bases do not exist for K n , n if n ≥ 8" /> <meta property="og:image" content="http://a.academia-assets.com/images/open-graph-icons/fb-paper.gif" /> <meta property="og:description" content="A basis for the cycle space of a graph is said to be robust if any cycle Z of G is a sum Z = C1 + C2 + · · · + Ck of basis elements such that (i) (C1 + C2 + · · · + Cl−1) ∩ Cl is a nontrivial path for each 2 ≤ l &lt; k. Hence, (ii) each partial sum" /> <meta property="article:author" content="https://georgetown.academia.edu/PaulKainen" /> <meta name="description" content="A basis for the cycle space of a graph is said to be robust if any cycle Z of G is a sum Z = C1 + C2 + · · · + Ck of basis elements such that (i) (C1 + C2 + · · · + Cl−1) ∩ Cl is a nontrivial path for each 2 ≤ l &lt; k. Hence, (ii) each partial sum" /> <title>(PDF) Robust cycle bases do not exist for K n , n if n ≥ 8 | Paul Kainen - Academia.edu</title> <link rel="canonical" href="https://www.academia.edu/65089431/Robust_cycle_bases_do_not_exist_for_K_n_n_if_n_8" /> <script async src="https://www.googletagmanager.com/gtag/js?id=G-5VKX33P2DS"></script> <script> window.dataLayer = window.dataLayer || []; function gtag(){dataLayer.push(arguments);} gtag('js', new Date()); gtag('config', 'G-5VKX33P2DS', { cookie_domain: 'academia.edu', send_page_view: false, }); gtag('event', 'page_view', { 'controller': "single_work", 'action': "show", 'controller_action': 'single_work#show', 'logged_in': 'false', 'edge': 'unknown', // Send nil if there is no A/B test bucket, in case some records get logged // with missing data - that way we can distinguish between the two cases. // ab_test_bucket should be of the form <ab_test_name>:<bucket> 'ab_test_bucket': null, }) </script> <script> var $controller_name = 'single_work'; var $action_name = "show"; var $rails_env = 'production'; var $app_rev = '92477ec68c09d28ae4730a4143c926f074776319'; var $domain = 'academia.edu'; var $app_host = "academia.edu"; var $asset_host = "academia-assets.com"; var $start_time = new Date().getTime(); var $recaptcha_key = "6LdxlRMTAAAAADnu_zyLhLg0YF9uACwz78shpjJB"; var $recaptcha_invisible_key = "6Lf3KHUUAAAAACggoMpmGJdQDtiyrjVlvGJ6BbAj"; var $disableClientRecordHit = false; </script> <script> window.require = { config: function() { return function() {} } } </script> <script> window.Aedu = window.Aedu || {}; window.Aedu.hit_data = null; window.Aedu.serverRenderTime = new Date(1732804479000); window.Aedu.timeDifference = new Date().getTime() - 1732804479000; </script> <script type="application/ld+json">{"@context":"https://schema.org","@type":"ScholarlyArticle","abstract":"A basis for the cycle space of a graph is said to be robust if any cycle Z of G is a sum Z = C1 + C2 + · · · + Ck of basis elements such that (i) (C1 + C2 + · · · + Cl−1) ∩ Cl is a nontrivial path for each 2 ≤ l \u0026amp;amp;lt; k. Hence, (ii) each partial sum C1 + C2 + · · ·+ Cl is a cycle for 1 ≤ l ≤ k. While complete graphs and 2-connected plane graphs have robust cycle bases, it is shown that regular complete bipartite graphs Kn,n do not have any robust cycle basis if n ≥ 8. © 2017 Elsevier B.V. All rights reserved.","author":[{"@context":"https://schema.org","@type":"Person","name":"Paul Kainen"}],"contributor":[],"dateCreated":"2021-12-19","dateModified":"2021-12-19","datePublished":"2017-01-01","headline":"Robust cycle bases do not exist for K n , n if n ≥ 8","inLanguage":"en","keywords":[],"locationCreated":null,"publication":null,"publisher":{"@context":"https://schema.org","@type":"Organization","name":null},"image":null,"thumbnailUrl":null,"url":"https://www.academia.edu/65089431/Robust_cycle_bases_do_not_exist_for_K_n_n_if_n_8","sourceOrganization":[{"@context":"https://schema.org","@type":"EducationalOrganization","name":"georgetown"}]}</script><link rel="stylesheet" media="all" href="//a.academia-assets.com/assets/single_work_page/loswp-102fa537001ba4d8dcd921ad9bd56c474abc201906ea4843e7e7efe9dfbf561d.css" /><link rel="stylesheet" media="all" href="//a.academia-assets.com/assets/design_system/body-8d679e925718b5e8e4b18e9a4fab37f7eaa99e43386459376559080ac8f2856a.css" 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Hence, (ii) each partial sum C1 + C2 + · · ·+ Cl is a cycle for 1 ≤ l ≤ k. While complete graphs and 2-connected plane graphs have robust cycle bases, it is shown that regular complete bipartite graphs Kn,n do not have any robust cycle basis if n ≥ 8. © 2017 Elsevier B.V. All rights reserved.","publication_date":"2017,,"},"document_type":"paper","pre_hit_view_count_baseline":null,"quality":"high","language":"en","title":"Robust cycle bases do not exist for K n , n if n ≥ 8","broadcastable":false,"draft":null,"has_indexable_attachment":true,"indexable":true}}["work"]; window.loswp.workCoauthors = [46765975]; window.loswp.locale = "en"; window.loswp.countryCode = "SG"; window.loswp.cwvAbTestBucket = ""; window.loswp.designVariant = "ds_vanilla"; window.loswp.fullPageMobileSutdModalVariant = "control"; window.loswp.useOptimizedScribd4genScript = false; window.loswp.appleClientId = 'edu.academia.applesignon';</script><script defer="" src="https://accounts.google.com/gsi/client"></script><div class="ds-loswp-container"><div class="ds-work-card--grid-container"><div class="ds-work-card--container js-loswp-work-card"><div class="ds-work-card--cover"><div class="ds-work-cover--wrapper"><div class="ds-work-cover--container"><button class="ds-work-cover--clickable js-swp-download-button" data-signup-modal="{"location":"swp-splash-paper-cover","attachmentId":76829438,"attachmentType":"pdf"}"><img alt="First page of “Robust cycle bases do not exist for K n , n if n ≥ 8”" class="ds-work-cover--cover-thumbnail" src="https://0.academia-photos.com/attachment_thumbnails/76829438/mini_magick20211219-21653-k93zba.png?1639919823" /><img alt="PDF Icon" class="ds-work-cover--file-icon" src="//a.academia-assets.com/assets/single_work_splash/adobe.icon-574afd46eb6b03a77a153a647fb47e30546f9215c0ee6a25df597a779717f9ef.svg" /><div class="ds-work-cover--hover-container"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span><p>Download Free PDF</p></div><div class="ds-work-cover--ribbon-container">Download Free PDF</div><div class="ds-work-cover--ribbon-triangle"></div></button></div></div></div><div class="ds-work-card--work-information"><h1 class="ds-work-card--work-title">Robust cycle bases do not exist for K n , n if n ≥ 8</h1><div class="ds-work-card--work-authors ds-work-card--detail"><a class="ds-work-card--author js-wsj-grid-card-author ds2-5-body-md ds2-5-body-link" data-author-id="46765975" href="https://georgetown.academia.edu/PaulKainen"><img alt="Profile image of Paul Kainen" class="ds-work-card--author-avatar" src="//a.academia-assets.com/images/s65_no_pic.png" />Paul Kainen</a></div><div class="ds-work-card--detail"><p class="ds-work-card--detail ds2-5-body-sm">2017</p></div><p class="ds-work-card--work-abstract ds-work-card--detail ds2-5-body-md">A basis for the cycle space of a graph is said to be robust if any cycle Z of G is a sum Z = C1 + C2 + · · · + Ck of basis elements such that (i) (C1 + C2 + · · · + Cl−1) ∩ Cl is a nontrivial path for each 2 ≤ l &lt; k. Hence, (ii) each partial sum C1 + C2 + · · ·+ Cl is a cycle for 1 ≤ l ≤ k. While complete graphs and 2-connected plane graphs have robust cycle bases, it is shown that regular complete bipartite graphs Kn,n do not have any robust cycle basis if n ≥ 8. © 2017 Elsevier B.V. All rights reserved.</p><div class="ds-work-card--button-container"><button class="ds2-5-button js-swp-download-button" data-signup-modal="{"location":"continue-reading-button--work-card","attachmentId":76829438,"attachmentType":"pdf","workUrl":"https://www.academia.edu/65089431/Robust_cycle_bases_do_not_exist_for_K_n_n_if_n_8"}">See full PDF</button><button class="ds2-5-button ds2-5-button--secondary js-swp-download-button" data-signup-modal="{"location":"download-pdf-button--work-card","attachmentId":76829438,"attachmentType":"pdf","workUrl":"https://www.academia.edu/65089431/Robust_cycle_bases_do_not_exist_for_K_n_n_if_n_8"}"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span>Download PDF</button></div></div></div></div><div data-auto_select="false" data-client_id="331998490334-rsn3chp12mbkiqhl6e7lu2q0mlbu0f1b" data-doc_id="76829438" data-landing_url="https://www.academia.edu/65089431/Robust_cycle_bases_do_not_exist_for_K_n_n_if_n_8" data-login_uri="https://www.academia.edu/registrations/google_one_tap" data-moment_callback="onGoogleOneTapEvent" id="g_id_onload"></div><div class="ds-top-related-works--grid-container"><div class="ds-related-content--container ds-top-related-works--container"><h2 class="ds-related-content--heading">Related papers</h2><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="0" data-entity-id="38176445" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/38176445/robust_cycle_bases_do_not_exist_for_Knn_if_n_geq_8_DAM_2018_pdf">robust-cycle-bases-do-not-exist-for-Knn-if-n-geq-8-DAM-2018.pdf</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="46765975" href="https://georgetown.academia.edu/PaulKainen">Paul Kainen</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Discrete Applied Math, 2018</p><p class="ds-related-work--abstract ds2-5-body-sm">A basis for the cycle space of a graph is said to be robust if any cycle Z of G is a sum Z = C1+C2+···+Ck of basis elements such that (i) (C1+C2+···+Cℓ−1)∩Cℓ is a nontrivial path for each 2 ≤ ℓ < k. Hence,(ii) each partial sum C1+C2+···+Cℓ is a cycle for 1≤ℓ≤k. While complete graphs and 2-connected plane graphs have robust cycle bases, it is shown that regular complete bipartite graphs Knn do not have any robust cycle basis if n≥8.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"robust-cycle-bases-do-not-exist-for-Knn-if-n-geq-8-DAM-2018.pdf","attachmentId":58212287,"attachmentType":"pdf","work_url":"https://www.academia.edu/38176445/robust_cycle_bases_do_not_exist_for_Knn_if_n_geq_8_DAM_2018_pdf","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/38176445/robust_cycle_bases_do_not_exist_for_Knn_if_n_geq_8_DAM_2018_pdf"><span 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class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="54259296" href="https://independent.academia.edu/TUeckerdt">T. Ueckerdt</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Computer Science Review, 2009</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Cycle bases in graphs characterization, algorithms, complexity, and applications","attachmentId":39153417,"attachmentType":"pdf","work_url":"https://www.academia.edu/16734782/Cycle_bases_in_graphs_characterization_algorithms_complexity_and_applications","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/16734782/Cycle_bases_in_graphs_characterization_algorithms_complexity_and_applications"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="4" data-entity-id="50044476" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/50044476/Independent_cycles_and_paths_in_bipartite_balanced_graphs">Independent cycles and paths in bipartite balanced graphs</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="56273669" href="https://independent.academia.edu/AdamPawe%C5%82Wojda">Adam Paweł Wojda</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Discussiones Mathematicae Graph Theory, 2008</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Independent cycles and paths in bipartite balanced graphs","attachmentId":68175358,"attachmentType":"pdf","work_url":"https://www.academia.edu/50044476/Independent_cycles_and_paths_in_bipartite_balanced_graphs","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/50044476/Independent_cycles_and_paths_in_bipartite_balanced_graphs"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="5" data-entity-id="56040640" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/56040640/On_certain_cycles_in_graphs">On certain cycles in graphs</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="44221456" href="https://independent.academia.edu/GrantDoug">Doug Grant</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Proceedings of the Edinburgh Mathematical Society, 1981</p><p class="ds-related-work--abstract ds2-5-body-sm">We show that every simple graph of order 2r and minimum degree ≧4r/3 has the property that for any partition of its vertex set into 2-subsets, there is a cycle which contains exactly one vertex from each 2-subset. We show that the bound 4r/3 cannot be lowered to r, but conjecture that it can be lowered to r + 1.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"On certain cycles in graphs","attachmentId":71622051,"attachmentType":"pdf","work_url":"https://www.academia.edu/56040640/On_certain_cycles_in_graphs","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/56040640/On_certain_cycles_in_graphs"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="6" data-entity-id="25824639" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/25824639/Convex_cycle_bases">Convex cycle bases</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="31551774" href="https://greifswald.academia.edu/MarcHellmuth">Marc Hellmuth</a><span>, </span><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="49706430" href="https://independent.academia.edu/JosefLeydold">Josef Leydold</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Ars Mathematica Contemporanea</p><p class="ds-related-work--abstract ds2-5-body-sm">Convex cycles play a role e.g. in the context of product graphs. We introduce convex cycle bases and describe a polynomial-time algorithm that recognizes whether a given graph has a convex cycle basis and provides an explicit construction in the positive case. Relations between convex cycles bases and other types of cycles bases are discussed. In particular we show that if G has a unique minimal cycle bases, this basis is convex. Furthermore, we characterize a class of graphs with convex cycles bases that includes partial cubes and hence median graphs.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Convex cycle bases","attachmentId":46191184,"attachmentType":"pdf","work_url":"https://www.academia.edu/25824639/Convex_cycle_bases","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/25824639/Convex_cycle_bases"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container 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