2026-07-22T04:12:15Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:16064 2024-03-06T10:56:52Z ddc:004 open_access

Disjointness Graphs of Short Polygonal Chains Pach, János Tardos, Gábor Tóth, Géza chi-bounded disjointness graph The disjointness graph of a set system is a graph whose vertices are the sets, two being connected by an edge if and only if they are disjoint. It is known that the disjointness graph G of any system of segments in the plane is χ-bounded, that is, its chromatic number χ(G) is upper bounded by a function of its clique number ω(G). Here we show that this statement does not remain true for systems of polygonal chains of length 2. We also construct systems of polygonal chains of length 3 such that their disjointness graphs have arbitrarily large girth and chromatic number. In the opposite direction, we show that the class of disjointness graphs of (possibly self-intersecting) 2-way infinite polygonal chains of length 3 is χ-bounded: for every such graph G, we have χ(G) ≤ (ω(G))³+ω(G). Schloss Dagstuhl – Leibniz-Zentrum für Informatik János Pach and Gábor Tardos and Géza Tóth 2022 Is Part Of LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.SoCG.2022.56 urn:nbn:de:0030-drops-160645 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.56 eng https://creativecommons.org/licenses/by/4.0/legalcode