eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
59:1
59:15
10.4230/LIPIcs.SoCG.2017.59
article
Disjointness Graphs of Segments
Pach, János
Tardos, Gábor
Tóth, Géza
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.59/LIPIcs.SoCG.2017.59.pdf
disjointness graph
chromatic number
clique number
chi-bounded