{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article9702","name":"Disjointness Graphs of Segments","abstract":"The disjointness graph G=G(S) of a set of segments S in R^d, d>1 is a graph whose vertex set is S and two vertices are connected by an edge if and only if the corresponding segments are disjoint. We prove that the chromatic number of G satisfies chi(G)<=omega(G)^4+omega(G)^3 where omega(G) denotes the clique number of G. It follows, that S has at least cn^{1\/5} pairwise intersecting or pairwise disjoint elements. Stronger bounds are established for lines in space, instead of segments.\r\n\r\nWe show that computing omega(G) and chi(G) for disjointness graphs of lines in space are NP-hard tasks. However, we can design efficient algorithms to compute proper colorings of G in which the number of colors satisfies the above upper bounds. One cannot expect similar results for sets of continuous arcs, instead of segments, even in the plane. We construct families of arcs whose disjointness graphs are triangle-free (omega(G)=2), but whose chromatic numbers are arbitrarily large.","keywords":["disjointness graph","chromatic number","clique number","chi-bounded"],"author":[{"@type":"Person","name":"Pach, J\u00e1nos","givenName":"J\u00e1nos","familyName":"Pach"},{"@type":"Person","name":"Tardos, G\u00e1bor","givenName":"G\u00e1bor","familyName":"Tardos"},{"@type":"Person","name":"T\u00f3th, G\u00e9za","givenName":"G\u00e9za","familyName":"T\u00f3th"}],"position":59,"pageStart":"59:1","pageEnd":"59:15","dateCreated":"2017-06-20","datePublished":"2017-06-20","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Pach, J\u00e1nos","givenName":"J\u00e1nos","familyName":"Pach"},{"@type":"Person","name":"Tardos, G\u00e1bor","givenName":"G\u00e1bor","familyName":"Tardos"},{"@type":"Person","name":"T\u00f3th, G\u00e9za","givenName":"G\u00e9za","familyName":"T\u00f3th"}],"copyrightYear":"2017","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SoCG.2017.59","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":{"@type":"PublicationVolume","@id":"#volume6280","volumeNumber":77,"name":"33rd International Symposium on Computational Geometry (SoCG 2017)","dateCreated":"2017-06-20","datePublished":"2017-06-20","editor":[{"@type":"Person","name":"Aronov, Boris","givenName":"Boris","familyName":"Aronov"},{"@type":"Person","name":"Katz, Matthew J.","givenName":"Matthew J.","familyName":"Katz"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article9702","isPartOf":{"@type":"Periodical","@id":"#series116","name":"Leibniz International Proceedings in Informatics","issn":"1868-8969","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume6280"}}}