Document Zbl 1284.05146

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J. Comb.</a> <a href="/?q=in%3A331041" title="Articles in this Issue">36, 503-530 (2014)</a>. </div> <div class="abstract">Summary: For a graph $G$, let $|G|$ denote its number of vertices, $\delta(G)$ its minimum degree and $\mathbb Z_1(G;\mathbb F_2)$ its cycle space. Call a graph Hamilton-generated if and only if every cycle in $G$ is a symmetric difference of some Hamilton circuits of $G$.<br class="zbmathjax-paragraph"> The main purpose of this paper is to prove: for every $\gamma>0$ there exists $n_0\in \mathbb Z$ such that for every graph $G$ with $|G|\geq n_0$ vertices,<br class="zbmathjax-paragraph"> (1) if $\delta(G)\geq (\frac{1}{2}+\gamma)|G|$ and $|G|$ is odd, then $G$ is Hamilton-generated,<br class="zbmathjax-paragraph"> (2) if $\delta(G)\geq (\frac{1}{2}+\gamma)|G|$ and $|G|$ is even, then the set of all Hamilton circuits of $G$ generates a codimension-one subspace of $\mathbb Z_1(G;\mathbb F_2)$ and the set of all circuits of $G$ having length either $|G|-1$ or $|G|$ generates all of $\mathbb Z_1(G;\mathbb F_2)$,<br class="zbmathjax-paragraph"> (3) if $\delta(G)\geq (\frac{1}{4}+\gamma)|G|$ and $G$ is balanced bipartite, then $G$ is Hamilton-generated.<br class="zbmathjax-paragraph"> All these degree-conditions are essentially best-possible. The implications in (1) and (2) give an asymptotic affirmative answer to a special case of an open conjecture which according to [<span class="zbmathjax-textit">I. B.-A. Hartman</span>, Eur. J. Comb. 4, 237–246 (1983; <a href="/0521.05039">Zbl 0521.05039</a>)] originates with A. Bondy.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A6273652">Cited in <strong>1</strong> Document</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A05C38" title="MSC2020">05C38</a> </td> <td class="space"> Paths and cycles </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A05C40" title="MSC2020">05C40</a> </td> <td class="space"> Connectivity </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Acycle+space">cycle space</a>; <a href="/?q=ut%3AEulerian+subgraphs">Eulerian subgraphs</a>; <a href="/?q=ut%3A2-connected+graph">2-connected graph</a></div><div class="keywords"> <h3>Citations:</h3><a href="/0521.05039">Zbl 0521.05039</a></div> <div class="software"> <h3>Software:</h3><a href="/software/6693">House of Graphs</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 1284.05146" data-ciurl="/ci/06273652" data-biburl="/bibtex/06273652.bib" data-amsurl="/amsrefs/06273652.bib" data-xmlurl="/xml/06273652.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/06273652.pdf" title="Zbl 1284.05146 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.ejc.2013.09.005" aria-label="DOI for “On prisms, Möbius ladders and the cycle space of dense graphs”" title="10.1016/j.ejc.2013.09.005">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/1112.5101"title="Note: arXiv document may differ from published version">arXiv</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Abdollahi, Alireza; Vatandoost, Ebrahim, Integral quartic Cayley graphs on abelian groups, Electron. 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