Document Zbl 1519.05068

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Geom.</a> <a href="/?q=in%3A500350" title="Articles in this Issue">115, Article ID 102020, 17 p. (2023)</a>. </div> <div class="abstract">Summary: We study problems related to colouring families of bottomless rectangles in the plane, in an attempt to improve the polychromatic $k$-colouring number $m_k^\ast$. This number is the smallest $m$ such that any collection of bottomless rectangles can be $k$-coloured so that any $m$-fold covered point is covered by all $k$ colours. We show that for many families of bottomless rectangles, such as unit-width bottomless rectangles, or bottomless rectangles whose left corners lie on a line, $m_k^\ast$ is linear in $k$. We present the lower bound $m_k^\ast \geq 2k - 1$ for general families.<br class="zbmathjax-paragraph">We also investigate semi-online colouring algorithms, which need not colour each vertex immediately, but must maintain a proper colouring. We prove that for many sweeping orders, for any positive integers $m, k$, there is no semi-online algorithm that can $k$-colour bottomless rectangles presented in that order, so that any $m$-fold covered point is covered by at least two colours. This holds even for translates of quadrants, and is a corollary of a stronger result for arborescence colourings: Any semi-online colouring algorithm that colours an arborescence presented in post-order may produce arbitrarily long monochromatic paths.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7720608">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%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%3A05C70" title="MSC2020">05C70</a> </td> <td class="space"> Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A52C15" title="MSC2020">52C15</a> </td> <td class="space"> Packing and covering in $2$ dimensions (aspects of discrete geometry) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A68W05" title="MSC2020">68W05</a> </td> <td class="space"> Nonnumerical algorithms </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A05C85" title="MSC2020">05C85</a> </td> <td class="space"> Graph algorithms (graph-theoretic aspects) </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Acombinatorial+geometry">combinatorial geometry</a>; <a href="/?q=ut%3Apolychromatic+colouring">polychromatic colouring</a>; <a href="/?q=ut%3Acover-decomposition">cover-decomposition</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 1519.05068" data-ciurl="/ci/07720608" data-biburl="/bibtex/07720608.bib" data-amsurl="/amsrefs/07720608.bib" data-xmlurl="/xml/07720608.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07720608.pdf" title="Zbl 1519.05068 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.comgeo.2023.102020" aria-label="DOI for “Colouring bottomless rectangles and arborescences”" title="10.1016/j.comgeo.2023.102020">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/1912.05251"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">Ackerman, E.; Keszegh, B.; Vizer, M., Colouring points with respect to squares, Discrete Comput. 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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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