Document Zbl 1072.68082

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Comput. Sci.</a> <a href="/?q=in%3A104066" title="Articles in this Issue">314, No. 3, 431-443 (2004)</a>. </div> <div class="abstract">Summary: The $n$-dimensional hypercube $Q_n$ is a graph having $2^n$ vertices labeled from 0 to $2^n-1$. Two vertices are connected by an edge if their binary labels differ in exactly one bit position. In this paper, we consider the faulty hypercube $Q_n$ with $n \geqslant 3$ that each vertex of $Q_n$ is incident to at least two nonfaulty edges. Based on this requirement, we prove that $Q_n$ contains a Hamiltonian path joining any two different colored vertices even if it has up to $2^n-5$ edge faults. Moreover, we show that there exists a path of length $2n-2$ between any two the same colored vertices in this faulty $Q_n$. Furthermore, we also prove that the faulty $Q_n$ still contains a cycle of every even length from 4 to $2^n$ inclusive.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A2081686">Cited in <strong>1</strong> Review</a></div><div class="clear"><a href="/?q=rf%3A2081686">Cited in <strong>58</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A68R10" title="MSC2020">68R10</a> </td> <td class="space"> Graph theory (including graph drawing) in computer science </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A68M15" title="MSC2020">68M15</a> </td> <td class="space"> Reliability, testing and fault tolerance of networks and computer systems </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3AHypercube">Hypercube</a>; <a href="/?q=ut%3AConditional+bipancyclic">Conditional bipancyclic</a>; <a href="/?q=ut%3AConditional+strongly+Hamiltonian+laceable">Conditional strongly Hamiltonian laceable</a>; <a href="/?q=ut%3AInterconnection+networks">Interconnection networks</a>; <a href="/?q=ut%3AFault-tolerant">Fault-tolerant</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1072.68082" data-ciurl="/ci/02081686" data-biburl="/bibtex/02081686.bib" data-amsurl="/amsrefs/02081686.bib" data-xmlurl="/xml/02081686.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/02081686.pdf" title="Zbl 1072.68082 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1016/j.tcs.2004.01.035" aria-label="DOI for “Linear array and ring embeddings in conditional faulty hypercubes”" title="10.1016/j.tcs.2004.01.035">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Akl, S. G., Parallel Computation: Models and Methods (1997), Prentice Hall: Prentice Hall NJ</td> </tr><tr> <td>[2]</td> <td class="space">Chan, M. Y.; Lee, S. J., On the existence of hamiltonian circuits in faulty hypercubes, SIAM J. Discrete Math., 4, 511-527 (1991) · <a href="/0747.05035" class="nowrap">Zbl 0747.05035</a></td> </tr><tr> <td>[3]</td> <td class="space">Harary, F.; Hayes, J. P., Edge fault tolerance in graphs, Networks, 23, 135-142 (1993) · <a href="/0796.05046" class="nowrap">Zbl 0796.05046</a></td> </tr><tr> <td>[4]</td> <td class="space">Hsieh, S. Y.; Chen, G. H.; Ho, C. W., Hamiltonian-laceability of star graphs, Networks, 36, 225-232 (2000) · <a href="/0968.05051" class="nowrap">Zbl 0968.05051</a></td> </tr><tr> <td>[5]</td> <td class="space">S. Latifi, S.Q. Zheng, N. Bagherzadeh, Optimal ring embedding in hypercubes with faulty links, in: Proc. of the IEEE Symp. on Fault-Tolerant Computing, 1992, pp. 178-184.; S. Latifi, S.Q. Zheng, N. Bagherzadeh, Optimal ring embedding in hypercubes with faulty links, in: Proc. of the IEEE Symp. on Fault-Tolerant Computing, 1992, pp. 178-184.</td> </tr><tr> <td>[6]</td> <td class="space">Leighton, F. T., Introduction to Parallel Algorithms and Architecture: Arrays, Trees, Hypercubes (1992), Morgan Kaufmann: Morgan Kaufmann San Mateo · <a href="/0743.68007" class="nowrap">Zbl 0743.68007</a></td> </tr><tr> <td>[7]</td> <td class="space">Lewinter, M.; Widulski, W., Hyper-hamilton laceable and caterpillar-spannable product graphs, Comput. Math. Appl., 34, 99-104 (1997) · <a href="/0907.05033" class="nowrap">Zbl 0907.05033</a></td> </tr><tr> <td>[8]</td> <td class="space">T.K. Li, C.H. Tsai, Jimmy J.M. Tan, L.H. Hsu, Bipanconnected and edge-fault-tolerant bipancyclic of hypercubes, Inform. Process. Lett. 87 (2003) 107-110.; T.K. Li, C.H. Tsai, Jimmy J.M. Tan, L.H. Hsu, Bipanconnected and edge-fault-tolerant bipancyclic of hypercubes, Inform. Process. Lett. 87 (2003) 107-110. · <a href="/1161.68684" class="nowrap">Zbl 1161.68684</a></td> </tr><tr> <td>[9]</td> <td class="space">Saad, Y.; Schultz, M. H., Topological properties of hypercubes, IEEE Trans. Comput., 37, 867-872 (1988)</td> </tr><tr> <td>[10]</td> <td class="space">Simmons, G., Almost all $n$-dimensional rectangular lattices are Hamilton laceable, Congr. Numer., 21, 103-108 (1978) · <a href="/0413.05029" class="nowrap">Zbl 0413.05029</a></td> </tr><tr> <td>[11]</td> <td class="space">C.H. Tsai, Jimmy J.M. Tan, T. Liang, L.H. Hsu, Fault tolerant hamiltonian laceability of hypercubes, Inform. Process. Lett. 83 (2002) 301-306.; C.H. Tsai, Jimmy J.M. Tan, T. Liang, L.H. Hsu, Fault tolerant hamiltonian laceability of hypercubes, Inform. Process. Lett. 83 (2002) 301-306. · <a href="/1043.68081" class="nowrap">Zbl 1043.68081</a></td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching. </div> </div></article> </div></div> </div> </div> <div class="clearfix"></div> </div> </div> <div id="foot"><div class="copyright"> © 2025 <a target="fiz" href="https://www.fiz-karlsruhe.de/en">FIZ Karlsruhe GmbH</a> <a href="/privacy-policy/">Privacy Policy</a> <a href="/legal-notices/">Legal Notices</a> <a href="/terms-conditions/">Terms & Conditions</a> <div class="info"> <ul class="nav"> <li class="mastodon"> <a href="https://mathstodon.xyz/@zbMATH" target="_blank" class="no-new-tab-icon"> <img src="/static/mastodon.png" title="zbMATH at Mathstodon (opens in new tab)" alt="Mastodon logo"> </a> </li> </ul> </div> </div> <div class="clearfix" style="height: 0px;"></div> </div> </div> <script src="https://static.zbmath.org/contrib/jquery/1.9.1/jquery.min.js"></script> <script src="https://static.zbmath.org/contrib/jquery-caret/1.5.2/jquery.caret.min.js"></script> <script src="/static/js/jquery-ui-1.10.1.custom.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap/v3.3.7zb1/js/bootstrap.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.js"></script> <script src="https://static.zbmath.org/contrib/retina/unknown/retina.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/js/bootstrap-select.min.js"></script> <script> var SCRIPT_ROOT = ""; </script> <script src="/static/scripts.js?v=20240926"> </script> <script src="https://static.zbmath.org/contrib/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script> <script type="text/x-mathjax-config"> MathJax.Hub.Config({ "HTML-CSS": { preferredFont: "TeX", availableFonts: [ "STIX", "TeX" ], linebreaks: { automatic: true }, EqnChunk: (MathJax.Hub.Browser.isMobile ? 10 : 50) }, tex2jax: { processEscapes: true, ignoreClass: "tex2jax_ignore|dno" }, TeX: { Macros: { Aut: "\\operatorname{Aut}", Hom: "\\operatorname{Hom}" }, noUndefined: { attributes: { mathcolor: "#039", //"red", mathbackground: "white", //"#FFEEEE", mathsize: "90%" } } }, messageStyle: "none" }); </script> <script type="text/javascript"> $(document).ready(function() { $("#MathInput").stop(true, true).keyup(function() { $.ajax({ url: "/mwsq/", type: "POST", data: { query : $("#MathInput").val() }, dataType: "text" }) .done(function(xml) { $("#MathPreview").html(xml); $(window).resize(); }); }); var press = jQuery.Event("keyup"); press.ctrlKey = false; press.which = 40; $("#MathInput").trigger(press); }); </script> <div id="new_tab_icon" style="display: none"> <span class="glyphicon glyphicon-new-window" aria-hidden="true"></span><span class="sr-only">(opens in new tab)</span></div> </body> </html>

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