Document Zbl 1244.05087

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Algorithms</a> <a href="/?q=in%3A301904" title="Articles in this Issue">40, No. 4, 425-436 (2012)</a>. </div> <div class="abstract">Summary: Let $G$ be a graph on $n$ vertices with maximum degree $\Delta $. We use the Lovász local lemma to show the following two results about colourings $\chi $ of the edges of the complete graph $K_{n}$. If for each vertex $v$ of $K_{n}$ the colouring $\chi $ assigns each colour to at most $(n - 2)/(22.4\Delta ^{2})$ edges emanating from $v$, then there is a copy of $G$ in $K_{n}$ which is properly edge-coloured by $\chi $. This improves on a result of <span class="zbmathjax-textit">N. Alon</span> et al. [Random Struct. Algorithms 23, No. 4, 409–433 (2003; <a href="/1037.05033">Zbl 1037.05033</a>)]. On the other hand, if $\chi $ assigns each colour to at most $n$/($51\Delta ^{2}$) edges of $K_{n}$, then there is a copy of $G$ in $K_{n}$ such that each edge of $G$ receives a different colour from $\chi $. This proves a conjecture of <span class="zbmathjax-textit">A. Frieze</span> and <span class="zbmathjax-textit">M. Krivelevich</span> [Electron. J. Comb. 15, No. 1, Research Paper R59, 9 p. (2008; <a href="/1159.05019">Zbl 1159.05019</a>)].<br class="zbmathjax-paragraph"> Our proofs rely on a framework developed by <span class="zbmathjax-textit">L. Lu</span> and <span class="zbmathjax-textit">L. Székely</span> [Electron. J. Comb. 14, No. 1, Research Paper R63, 13 p. (2007; <a href="/1183.05088">Zbl 1183.05088</a>)] for applying the local lemma to random injections. In order to improve the constants in our results we use a version of the local lemma due to <span class="zbmathjax-textit">R. Bissacot</span> et al. [Comb. Probab. Comput. 20, No. 5, 709–719 (2011; <a href="/1233.05196">Zbl 1233.05196</a>)].</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A6065785">Cited in <strong>23</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A05C15" title="MSC2020">05C15</a> </td> <td class="space"> Coloring of graphs and hypergraphs </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A05C07" title="MSC2020">05C07</a> </td> <td class="space"> Vertex degrees </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A05D10" title="MSC2020">05D10</a> </td> <td class="space"> Ramsey theory </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3ARamsey+theory">Ramsey theory</a>; <a href="/?q=ut%3ALovasz+local+lemma">Lovasz local lemma</a>; <a href="/?q=ut%3Arainbow+colourings">rainbow colourings</a>; <a href="/?q=ut%3Aproper+edge+colourings">proper edge colourings</a></div><div class="keywords"> <h3>Citations:</h3><a href="/1037.05033">Zbl 1037.05033</a>; <a href="/1159.05019">Zbl 1159.05019</a>; <a href="/1183.05088">Zbl 1183.05088</a>; <a href="/1233.05196">Zbl 1233.05196</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 1244.05087" data-ciurl="/ci/06065785" data-biburl="/bibtex/06065785.bib" data-amsurl="/amsrefs/06065785.bib" data-xmlurl="/xml/06065785.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/06065785.pdf" title="Zbl 1244.05087 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/10.1002/rsa.20383" aria-label="DOI for “Properly coloured copies and rainbow copies of large graphs with small maximum degree”" title="10.1002/rsa.20383">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/1007.3767"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">Albert, Multicoloured Hamilton cycles, Electron J Comb 2 (1995) · <a href="/0817.05028" class="nowrap">Zbl 0817.05028</a></td> </tr><tr> <td>[2]</td> <td class="space">Alon, Properly colored Hamilton cycles in edge-colored complete graphs, Random Struct Algorithms 11 pp 179– (1997) · <a href="/0882.05084" class="nowrap">Zbl 0882.05084</a> · <a href="https://doi.org/10.1002/(SICI)1098-2418(199709)11:2<179::AID-RSA5>3.0.CO;2-P" class="nowrap">doi:10.1002/(SICI)1098-2418(199709)11:2<179::AID-RSA5>3.0.CO;2-P</a></td> </tr><tr> <td>[3]</td> <td class="space">Alon, Properly colored subgraphs and rainbow subgraphs in edge-colorings with local constraints, Random Struct Algorithms 23 pp 409– (2003) · <a href="/1037.05033" class="nowrap">Zbl 1037.05033</a> · <a href="https://doi.org/10.1002/rsa.10102" class="nowrap">doi:10.1002/rsa.10102</a></td> </tr><tr> <td>[4]</td> <td class="space">Bissacot, An improvement of the Lovász local lemma via cluster expansion, Comb Probab Comput</td> </tr><tr> <td>[5]</td> <td class="space">Bollobás, Alternating Hamiltonian cycles, Isr J Math 23 pp 126– (1976) · <a href="/0325.05114" class="nowrap">Zbl 0325.05114</a> · <a href="https://doi.org/10.1007/BF02756791" class="nowrap">doi:10.1007/BF02756791</a></td> </tr><tr> <td>[6]</td> <td class="space">J. Böttcher Y. Kohayakawa A. Taraz Almost spanning subgraphs of random graphs after adversarial edge removal · <a href="/1268.05176" class="nowrap">Zbl 1268.05176</a></td> </tr><tr> <td>[7]</td> <td class="space">Chen, Graphs with Hamiltonian cycles having adjacent lines different colors, J Combin Theory Ser B 21 pp 135– (1976) · <a href="/0287.05106" class="nowrap">Zbl 0287.05106</a> · <a href="https://doi.org/10.1016/0095-8956(76)90053-8" class="nowrap">doi:10.1016/0095-8956(76)90053-8</a></td> </tr><tr> <td>[8]</td> <td class="space">Erdos, Lopsided Lovász local lemma and Latin transversals, Discrete Appl Math 30 pp 151– (1991) · <a href="/0717.05017" class="nowrap">Zbl 0717.05017</a> · <a href="https://doi.org/10.1016/0166-218X(91)90040-4" class="nowrap">doi:10.1016/0166-218X(91)90040-4</a></td> </tr><tr> <td>[9]</td> <td class="space">Erdos, Colloq. Math. Soc. Janos Bolyai 10, in: Infinite finite Sets, Colloq. Honour Paul Erdos, Keszthely 1973 pp 609– (1975)</td> </tr><tr> <td>[10]</td> <td class="space">Erdos, Teubner-Texte Math., in: Graphs and other combinatorial topics (Prague, 1982) pp 54– (1983)</td> </tr><tr> <td>[11]</td> <td class="space">Erdos, A combinatorial theorem, J London Math Soc 25 pp 249– (1950) · <a href="/0038.15301" class="nowrap">Zbl 0038.15301</a> · <a href="https://doi.org/10.1112/jlms/s1-25.4.249" class="nowrap">doi:10.1112/jlms/s1-25.4.249</a></td> </tr><tr> <td>[12]</td> <td class="space">Frieze, On rainbow trees and cycles, Electron J Comb 15 (2008) · <a href="/1159.05019" class="nowrap">Zbl 1159.05019</a></td> </tr><tr> <td>[13]</td> <td class="space">Frieze, Polychromatic Hamilton cycles, Discrete Math 118 pp 69– (1993) · <a href="/0803.05036" class="nowrap">Zbl 0803.05036</a> · <a href="https://doi.org/10.1016/0012-365X(93)90054-W" class="nowrap">doi:10.1016/0012-365X(93)90054-W</a></td> </tr><tr> <td>[14]</td> <td class="space">Hahn, Path and cycle sub-Ramsey numbers and an edge-colouring conjecture, Discrete Math 62 pp 29– (1986) · <a href="/0613.05044" class="nowrap">Zbl 0613.05044</a> · <a href="https://doi.org/10.1016/0012-365X(86)90038-5" class="nowrap">doi:10.1016/0012-365X(86)90038-5</a></td> </tr><tr> <td>[15]</td> <td class="space">Lu, Using Lovász local lemma in the space of random injections, Electron J Comb 14 (2007)</td> </tr><tr> <td>[16]</td> <td class="space">Ramsey, On a problem of formal logic, Proc London Math Soc 30 pp 264– (1930) · <a href="/55.0032.04" class="nowrap">JFM 55.0032.04</a> · <a href="https://doi.org/10.1112/plms/s2-30.1.264" class="nowrap">doi:10.1112/plms/s2-30.1.264</a></td> </tr><tr> <td>[17]</td> <td class="space">Rue, Comment on [1], Electron J Comb 2 (1995)</td> </tr><tr> <td>[18]</td> <td class="space">Shearer, A property of the colored complete graph, Discrete Math 25 pp 175– (1979) · <a href="/0397.05024" class="nowrap">Zbl 0397.05024</a> · <a href="https://doi.org/10.1016/0012-365X(79)90020-7" class="nowrap">doi:10.1016/0012-365X(79)90020-7</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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