2026-07-20T06:00:11Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:7196 2024-03-06T10:40:04Z ddc:004 open_access

Disjointness Graphs of Segments Pach, János Tardos, Gábor Tóth, Géza disjointness graph chromatic number clique number chi-bounded 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. We 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik János Pach and Gábor Tardos and Géza Tóth 2017 Is Part Of LIPIcs, Volume 77, 33rd International Symposium on Computational Geometry (SoCG 2017) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.SoCG.2017.59 urn:nbn:de:0030-drops-71960 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2017.59 eng https://creativecommons.org/licenses/by/3.0/legalcode