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Hence,(ii) each partial sum C1+C2+···+Cℓ is a" /> <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/38176445/robust_cycle_bases_do_not_exist_for_Knn_if_n_geq_8_DAM_2018_pdf" /> <meta property="og:title" content="robust-cycle-bases-do-not-exist-for-Knn-if-n-geq-8-DAM-2018.pdf" /> <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 C1+C2+···+Cℓ is a" /> <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 C1+C2+···+Cℓ is a" /> <title>(PDF) robust-cycle-bases-do-not-exist-for-Knn-if-n-geq-8-DAM-2018.pdf</title> <link rel="canonical" href="https://www.academia.edu/38176445/robust_cycle_bases_do_not_exist_for_Knn_if_n_geq_8_DAM_2018_pdf" /> <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 = '1e60a92a442ff83025cbe4f252857ee7c49c0bbe'; 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(1740567263000); window.Aedu.timeDifference = new Date().getTime() - 1740567263000; </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;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 Knn do not have any robust cycle basis if n≥8.","author":[{"@context":"https://schema.org","@type":"Person","name":"Paul Kainen","url":"https://georgetown.academia.edu/PaulKainen"}],"contributor":[],"dateCreated":"2019-01-18","dateModified":"2024-11-26","datePublished":"2018-01-01","headline":"robust-cycle-bases-do-not-exist-for-Knn-if-n-geq-8-DAM-2018.pdf","identifier":{"@type":"PropertyValue","propertyID":"DOI","value":"10.1016/j.dam.2017.10.001 0166-218X"},"image":"https://attachments.academia-assets.com/58212287/thumbnails/1.jpg","inLanguage":"en","keywords":[],"publication":"Discrete Applied Math","publisher":{"@context":"https://schema.org","@type":"Organization","name":null},"sameAs":"https://doi.org/10.1016/j.dam.2017.10.001 0166-218X","sourceOrganization":[{"@context":"https://schema.org","@type":"EducationalOrganization","name":"georgetown"}],"thumbnailUrl":"https://attachments.academia-assets.com/58212287/thumbnails/1.jpg","url":"https://www.academia.edu/38176445/robust_cycle_bases_do_not_exist_for_Knn_if_n_geq_8_DAM_2018_pdf"}</script><style type="text/css">@media(max-width: 567px){:root{--token-mode: Rebrand;--dropshadow: 0 2px 4px 0 #22223340;--primary-brand: #0645b1;--error-dark: #b60000;--success-dark: #05b01c;--inactive-fill: #ebebee;--hover: #0c3b8d;--pressed: #082f75;--button-primary-fill-inactive: #ebebee;--button-primary-fill: #0645b1;--button-primary-text: #ffffff;--button-primary-fill-hover: #0c3b8d;--button-primary-fill-press: #082f75;--button-primary-icon: #ffffff;--button-primary-fill-inverse: #ffffff;--button-primary-text-inverse: #082f75;--button-primary-icon-inverse: #0645b1;--button-primary-fill-inverse-hover: #cddaef;--button-primary-stroke-inverse-pressed: 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{"work":{"id":38176445,"created_at":"2019-01-18T12:31:15.114-08:00","from_world_paper_id":null,"updated_at":"2024-11-26T22:37:03.341-08:00","_data":{"doi":"10.1016/j.dam.2017.10.001 0166-218X","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 \u003c 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.","ai_title_tag":"Non-existence of Robust Cycle Bases in Knn for n≥8","publication_date":"2018,,","publication_name":"Discrete Applied Math"},"document_type":"paper","pre_hit_view_count_baseline":null,"quality":"high","language":"en","title":"robust-cycle-bases-do-not-exist-for-Knn-if-n-geq-8-DAM-2018.pdf","broadcastable":true,"draft":false,"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 = "full_page_mobile_sutd_modal"; window.loswp.useOptimizedScribd4genScript = false; window.loginModal = {}; window.loginModal.appleClientId = 'edu.academia.applesignon'; window.userInChina = "false";</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":58212287,"attachmentType":"pdf"}"><img alt="First page of “robust-cycle-bases-do-not-exist-for-Knn-if-n-geq-8-DAM-2018.pdf”" class="ds-work-cover--cover-thumbnail" src="https://0.academia-photos.com/attachment_thumbnails/58212287/mini_magick20190118-29191-5be90.png?1547850055" /><img alt="PDF Icon" class="ds-work-cover--file-icon" src="//a.academia-assets.com/images/single_work_splash/adobe_icon.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-Knn-if-n-geq-8-DAM-2018.pdf</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">2018, Discrete Applied Math</p><a class="js-loswp-work-card-doi-link ds2-5-body-sm ds2-5-body-link" href="https://doi.org/10.1016/j.dam.2017.10.001 0166-218X" rel="nofollow">https://doi.org/10.1016/j.dam.2017.10.001 0166-218X</a><div class="ds-work-card--work-metadata"><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">visibility</span><p class="ds2-5-body-sm" id="work-metadata-view-count">…</p></div><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">description</span><p class="ds2-5-body-sm">6 pages</p></div><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">link</span><p class="ds2-5-body-sm">1 file</p></div></div><script>(async () => { const workId = 38176445; 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if (!viewCountBody) { throw new Error('Failed to find work views element'); } viewCountBody.textContent = `${commaizedViewCount} views`; } catch (error) { // Remove the whole views element if there was some issue parsing. document.getElementById('work-metadata-view-count')?.parentNode?.remove(); throw new Error(`Failed to parse view count: ${viewCount}`, error); } }; // If the DOM is still loading, wait for it to be ready before updating the view count. if (document.readyState === "loading") { document.addEventListener('DOMContentLoaded', () => { updateViewCount(viewCount); }); // Otherwise, just update it immediately. } else { updateViewCount(viewCount); } })();</script></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 < 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-work-card--button-container"><button class="ds2-5-button js-swp-download-button" data-signup-modal="{"location":"continue-reading-button--work-card","attachmentId":58212287,"attachmentType":"pdf","workUrl":"https://www.academia.edu/38176445/robust_cycle_bases_do_not_exist_for_Knn_if_n_geq_8_DAM_2018_pdf"}">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":58212287,"attachmentType":"pdf","workUrl":"https://www.academia.edu/38176445/robust_cycle_bases_do_not_exist_for_Knn_if_n_geq_8_DAM_2018_pdf"}"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span>Download PDF</button></div><div class="ds-signup-banner-trigger-container"><div class="ds-signup-banner-trigger ds-signup-banner-trigger-control"></div></div><div class="ds-signup-banner ds-signup-banner-control"><div id="ds-signup-banner-close-button"><button class="ds2-5-button ds2-5-button--secondary ds2-5-button--inverse"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">close</span></button></div><div class="ds-signup-banner-ctas" data-impression-entity-id="38176445" data-impression-entity-type="2" data-impression-source="signup-banner"><img src="//a.academia-assets.com/images/academia-logo-capital-white.svg" /><h4 class="ds2-5-heading-serif-sm">Sign up for access to the world's latest research</h4><button class="ds2-5-button ds2-5-button--inverse ds2-5-button--full-width js-swp-download-button" data-signup-modal="{"location":"signup-banner"}">Sign up for free<span class="material-symbols-outlined" style="font-size: 20px" translate="no">arrow_forward</span></button></div><div class="ds-signup-banner-divider"></div><div class="ds-signup-banner-reasons"><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Get notified about relevant papers</span></div><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Save papers to use in your research</span></div><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Join the discussion with peers</span></div><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Track your impact</span></div></div></div><script>(() => { // Set up signup banner show/hide behavior: // 1. 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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.</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 K n , n if n ≥ 8","attachmentId":76829438,"attachmentType":"pdf","work_url":"https://www.academia.edu/65089431/Robust_cycle_bases_do_not_exist_for_K_n_n_if_n_8","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/65089431/Robust_cycle_bases_do_not_exist_for_K_n_n_if_n_8"><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="1" data-entity-id="12606806" 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/12606806/A_note_on_quasi_robust_cycle_bases">A note on quasi-robust 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></div><p class="ds-related-work--metadata ds2-5-body-xs">2009</p><p class="ds-related-work--abstract ds2-5-body-sm">We investigate here some aspects of cycle bases of undirected graphs that allow the iterative construction of all elementary cycles. We introduce the concept of quasi-robust bases as a generalization of the notion of robust bases and demonstrate that a certain class of bases of the complete bipartite graphs K m,n with m, n ≥ 5 is quasi-robust but not robust. We furthermore disprove a conjecture for cycle bases of Cartesian product 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":"A note on quasi-robust cycle bases","attachmentId":37742106,"attachmentType":"pdf","work_url":"https://www.academia.edu/12606806/A_note_on_quasi_robust_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/12606806/A_note_on_quasi_robust_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 js-wsj-grid-card" data-collection-position="2" data-entity-id="57257418" 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/57257418/On_robust_cycle_bases">On robust 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="46765975" href="https://georgetown.academia.edu/PaulKainen">Paul Kainen</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Electronic Notes in Discrete Mathematics, 2002</p><p class="ds-related-work--abstract ds2-5-body-sm">Two types of robust cycle bases are defined via recursively nice arrangements; complete and bipartite complete graphs are shown to have such bases. It is shown that a diagram in a groupoid is commutative up to natural equivalence (cutne) if for each cycle in a robust basis of the graph underlying the diagram, the composition of the morphisms is naturally equivalent to the identity. For a hypercube Q n , it is shown that the commutativity (or cutne) of a particular subset of asymptotically 4/n of the square faces forces commutativity (or cutne) of the entire diagram.</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 robust cycle bases","attachmentId":72244317,"attachmentType":"pdf","work_url":"https://www.academia.edu/57257418/On_robust_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/57257418/On_robust_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 js-wsj-grid-card" data-collection-position="3" data-entity-id="16734782" 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/16734782/Cycle_bases_in_graphs_characterization_algorithms_complexity_and_applications">Cycle bases in graphs characterization, algorithms, complexity, and applications</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="36140287" href="https://independent.academia.edu/MichailD">Dimitrios Michail</a><span>, </span><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="39133111" href="https://univr.academia.edu/RomeoRizzi">Romeo Rizzi</a><span>, </span><a 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><p class="ds-related-work--abstract ds2-5-body-sm">Cycles in graphs play an important role in many applications, e.g., analysis of electrical networks, analysis of chemical and biological pathways, periodic scheduling, and graph drawing. From a mathematical point of view, cycles in graphs have a rich structure. Cycle bases are a compact description of the set of all cycles of a graph. In this paper, we survey the state of knowledge on cycle bases and also derive some new results. We introduce different kinds of cycle bases, characterize them in terms of their cycle matrix, and prove structural results and apriori length bounds. We provide polynomial algorithms for the minimum cycle basis problem for some of the classes and prove APX -hardness for others. We also discuss three applications and show that they require different kinds of cycle bases.</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><p class="ds-related-work--abstract ds2-5-body-sm">We prove that if G and H are two bipartite balanced graphs of order |G| = |H| = 2p ≥ 4 such that the sizes of G and H satisfy G ≤ 2p − 3 and H ≤ 2p − 2, and the maximum degree of H is at most 2, then G and H are bi-placeable, unless G and H is one of easily recognizable couples of graphs. This result implies easily that for integers p and k 1 , k 2 ,. .. , k l such that k i ≥ 2 for i = 1,. .. , l and k 1 + • • • + k l ≤ p − 1 every bipartite balanced graph G of order 2p and size at least p 2 − 2p + 3 contains mutually vertex disjoint cycles C 2k1 ,. .. , C 2k l , unless G = K 3,3 −3K 1,1 .</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":"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="24881569" 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/24881569/The_basis_number_of_some_special_non_planar_graphs">The basis number of some special non-planar 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="47965953" href="https://independent.academia.edu/SalarAlsardary">Salar Alsardary</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Czechoslovak Mathematical Journal, 2003</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":"The basis number of some special non-planar graphs","attachmentId":45202625,"attachmentType":"pdf","work_url":"https://www.academia.edu/24881569/The_basis_number_of_some_special_non_planar_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/24881569/The_basis_number_of_some_special_non_planar_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="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="7" 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 js-wsj-grid-card" data-collection-position="8" data-entity-id="32671925" 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/32671925/Cycle_systems_in_the_complete_bipartite_graph_minus_a_one_factor">Cycle systems in the complete bipartite graph minus a one-factor</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="63531405" href="https://independent.academia.edu/JeffreyDinitz">Jeffrey Dinitz</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Discrete Mathematics, 2004</p><p class="ds-related-work--abstract ds2-5-body-sm">Let K n,n − I denote the complete bipartite graph with n vertices in each part from which a 1-factor I has been removed. An m-cycle system of K n,n − I is a collection of m-cycles whose edges partition K n,n − I. Necessary conditions for the existence of such an m-cycle system are that m ≥ 4 is even, n ≥ 3 is odd, m ≤ 2n, and m | n(n − 1). In this paper, we show these necessary conditions are sufficient except possibly in the case that m ≡ 0 (mod 4) with n < m < 2n.</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 systems in the complete bipartite graph minus a one-factor","attachmentId":52838346,"attachmentType":"pdf","work_url":"https://www.academia.edu/32671925/Cycle_systems_in_the_complete_bipartite_graph_minus_a_one_factor","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/32671925/Cycle_systems_in_the_complete_bipartite_graph_minus_a_one_factor"><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="9" data-entity-id="18984856" 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/18984856/New_length_bounds_for_cycle_bases">New length bounds for 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="39133111" href="https://univr.academia.edu/RomeoRizzi">Romeo Rizzi</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Information Processing Letters, 2007</p><p class="ds-related-work--abstract ds2-5-body-sm">Based on a recent work by Abraham, , we construct a strictly fundamental cycle basis of length O(n 2 ) for any unweighted graph, whence proving the conjecture of .</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":"New length bounds for cycle bases","attachmentId":40366679,"attachmentType":"pdf","work_url":"https://www.academia.edu/18984856/New_length_bounds_for_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/18984856/New_length_bounds_for_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></div><div class="ds-sticky-ctas--wrapper js-loswp-sticky-ctas hidden"><div class="ds-sticky-ctas--grid-container"><div class="ds-sticky-ctas--container"><button class="ds2-5-button js-swp-download-button" data-signup-modal="{"location":"continue-reading-button--sticky-ctas","attachmentId":58212287,"attachmentType":"pdf","workUrl":null}">See full PDF</button><button class="ds2-5-button ds2-5-button--secondary js-swp-download-button" data-signup-modal="{"location":"download-pdf-button--sticky-ctas","attachmentId":58212287,"attachmentType":"pdf","workUrl":null}"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span>Download PDF</button></div></div></div><div class="ds-below-fold--grid-container"><div class="ds-work--container js-loswp-embedded-document"><div class="attachment_preview" data-attachment="Attachment_58212287" style="display: none"><div class="js-scribd-document-container"><div class="scribd--document-loading js-scribd-document-loader" style="display: block;"><img alt="Loading..." src="//a.academia-assets.com/images/loaders/paper-load.gif" /><p>Loading Preview</p></div></div><div style="text-align: center;"><div class="scribd--no-preview-alert js-preview-unavailable"><p>Sorry, preview is currently unavailable. You can download the paper by clicking the button above.</p></div></div></div></div><div class="ds-sidebar--container js-work-sidebar"><div class="ds-related-content--container"><h2 class="ds-related-content--heading">Related papers</h2><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="0" data-entity-id="77130472" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/77130472/Hamiltonicity_Pancyclicity_and_Cycle_Extendability_in_Graphs">Hamiltonicity, Pancyclicity, and Cycle Extendability in Graphs</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="71484" href="https://hcc-nd.academia.edu/DeborahArangnoPhD">Deborah Arangno, PhD</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2014</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link 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data-entity-id="9736733" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/9736733/Balanced_cycles_and_holes_in_bipartite_graphs">Balanced cycles and holes in bipartite graphs</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="23405089" href="https://independent.academia.edu/KristinaVuskovic">Kristina Vuskovic</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Discrete Mathematics, 1999</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":"Balanced cycles and holes in bipartite graphs","attachmentId":47661911,"attachmentType":"pdf","work_url":"https://www.academia.edu/9736733/Balanced_cycles_and_holes_in_bipartite_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-related-work-grid-card-view-pdf" href="https://www.academia.edu/9736733/Balanced_cycles_and_holes_in_bipartite_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-related-work-sidebar-card" data-collection-position="2" data-entity-id="21885111" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/21885111/A_cycle_structure_theorem_for_hamiltonian_graphs">A cycle structure theorem for hamiltonian graphs</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm 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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-related-work-sidebar-card" data-collection-position="3" data-entity-id="105782510" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/105782510/Cycles_in_traceable_oriented_graphs">Cycles in -traceable oriented graphs</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="48759345" href="https://independent.academia.edu/MarietjieFrick">Marietjie Frick</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Discrete Mathematics, 2011</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" 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