Document Zbl 1248.05068

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Geom.</a> <a href="/?q=in%3A302414" title="Articles in this Issue">45, No. 9, 495-507 (2012)</a>. </div> <div class="abstract">Summary: We prove lower and upper bounds for the chromatic number of certain hypergraphs defined by geometric regions. This problem has close relations to conflict-free colorings. One of the most interesting type of regions to consider for this problem is that of the axis-parallel rectangles. We completely solve the problem for a special case of them, for bottomless rectangles. We also give an almost complete answer for half-planes and pose several open problems. Moreover, we give efficient coloring algorithms.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A6084814">Cited in <strong>11</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%3A05C65" title="MSC2020">05C65</a> </td> <td class="space"> Hypergraphs </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A05B25" title="MSC2020">05B25</a> </td> <td class="space"> Combinatorial aspects of finite geometries </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aproper+hypergraph+coloring">proper hypergraph coloring</a>; <a href="/?q=ut%3Ageometric+hypergraph">geometric hypergraph</a>; <a href="/?q=ut%3Aconflict-free+coloring">conflict-free coloring</a>; <a href="/?q=ut%3Ahalf-plane">half-plane</a>; <a href="/?q=ut%3Aaxis-parallel+rectangle">axis-parallel rectangle</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 1248.05068" data-ciurl="/ci/06084814" data-biburl="/bibtex/06084814.bib" data-amsurl="/amsrefs/06084814.bib" data-xmlurl="/xml/06084814.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/06084814.pdf" title="Zbl 1248.05068 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.2011.09.004" aria-label="DOI for “Coloring half-planes and bottomless rectangles”" title="10.1016/j.comgeo.2011.09.004">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/1105.0169"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">D. Ajwani, K. Elbassioni, S. Govindarajan, S. Ray, Conflict-free coloring for rectangle ranges using $\widetilde{O}( n^{. 382 + \epsilon})$; D. Ajwani, K. Elbassioni, S. Govindarajan, S. Ray, Conflict-free coloring for rectangle ranges using $\widetilde{O}( n^{. 382 + \epsilon})$ · <a href="/1246.68167" class="nowrap">Zbl 1246.68167</a></td> </tr><tr> <td>[2]</td> <td class="space">G. Brodal, R. Jacob, Dynamic planar convex hull, in: Proc. 43rd IEEE Symp. on Foundations of Computer Science, 2002, pp. 617-626.; G. Brodal, R. Jacob, Dynamic planar convex hull, in: Proc. 43rd IEEE Symp. on Foundations of Computer Science, 2002, pp. 617-626.</td> </tr><tr> <td>[3]</td> <td class="space">X. Chen, J. Pach, M. Szegedy, G. Tardos, Delaunay graphs of point sets in the plane with respect to axis-parallel rectangles, in: SODAʼ08: Proceedings of the Nineteenth Annual ACM-SIAM Symposium on Discrete Algorithms, 2008, pp. 94-101.; X. Chen, J. Pach, M. Szegedy, G. Tardos, Delaunay graphs of point sets in the plane with respect to axis-parallel rectangles, in: SODAʼ08: Proceedings of the Nineteenth Annual ACM-SIAM Symposium on Discrete Algorithms, 2008, pp. 94-101. · <a href="/1192.05099" class="nowrap">Zbl 1192.05099</a></td> </tr><tr> <td>[4]</td> <td class="space">Even, G.; Lotker, Z.; Ron, D.; Smorodinsky, S., Conflict-free colorings of simple geometric regions with applications to frequency assignment in cellular networks, SIAM J. Comput., 33, 1, 94-136 (2003) · <a href="/1069.68120" class="nowrap">Zbl 1069.68120</a></td> </tr><tr> <td>[5]</td> <td class="space">R. Fulek, Coloring geometric hypergraph defined by an arrangement of half-planes, in: Proceedings of the 22th Canadian Conference on Computational Geometry, 2010, pp. 71-74.; R. Fulek, Coloring geometric hypergraph defined by an arrangement of half-planes, in: Proceedings of the 22th Canadian Conference on Computational Geometry, 2010, pp. 71-74.</td> </tr><tr> <td>[6]</td> <td class="space">Har-Peled, S.; Smorodinsky, S., On conflict-free coloring of points and simple regions in the plane, Discrete Comput. Geom. (DCG), 34, 47-70 (2005) · <a href="/1066.05064" class="nowrap">Zbl 1066.05064</a></td> </tr><tr> <td>[7]</td> <td class="space">B. Keszegh, Weak conflict-free colorings of point sets and simple regions, in: Proceedings of the 19th Canadian Conference on Computational Geometry, 2007, pp. 97-100.; B. Keszegh, Weak conflict-free colorings of point sets and simple regions, in: Proceedings of the 19th Canadian Conference on Computational Geometry, 2007, pp. 97-100.</td> </tr><tr> <td>[8]</td> <td class="space">B. Keszegh, Combinatorial and computational problems about points in the plane, PhD Dissertation, Central European University, Department of Mathematics and its Applications, 2009.; B. Keszegh, Combinatorial and computational problems about points in the plane, PhD Dissertation, Central European University, Department of Mathematics and its Applications, 2009.</td> </tr><tr> <td>[9]</td> <td class="space">Pach, J.; Tardos, G., Coloring axis-parallel rectangles, J. Combin. Theory Ser. A, 117, 6, 776-782 (2010) · <a href="/1221.05160" class="nowrap">Zbl 1221.05160</a></td> </tr><tr> <td>[10]</td> <td class="space">Pach, J.; Tóth, G., Conflict free colorings, (Basu, S.; etal., Discrete and Computational Geometry - The Goodman-Pollack Festschrift (2003), Springer-Verlag: Springer-Verlag Berlin), 665-671 · <a href="/1104.68545" class="nowrap">Zbl 1104.68545</a></td> </tr><tr> <td>[11]</td> <td class="space">Pach, J.; Tardos, G.; Tóth, G., Indecomposable coverings, (Discrete Geometry, Combinatorics and Graph Theory. Discrete Geometry, Combinatorics and Graph Theory, Lecture Notes in Computer Science, vol. 4381 (2007), Springer), 135-148 · <a href="/1149.52304" class="nowrap">Zbl 1149.52304</a></td> </tr><tr> <td>[12]</td> <td class="space">S. Smorodinsky, Combinatorial problems in computational geometry, PhD thesis, School of Computer Science, Tel-Aviv University, Tel-Aviv, Israel, 2003.; S. Smorodinsky, Combinatorial problems in computational geometry, PhD thesis, School of Computer Science, Tel-Aviv University, Tel-Aviv, Israel, 2003.</td> </tr><tr> <td>[13]</td> <td class="space">Smorodinsky, S., On the chromatic number of some geometric hypergraphs, SIAM J. Discrete Math., 21, 3, 676-687 (2007) · <a href="/1141.05064" class="nowrap">Zbl 1141.05064</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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