Disjointness Graphs of Short Polygonal Chains

Authors János Pach, Gábor Tardos, Géza Tóth

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János Pach
  • Rényi Institute, Budapest, Hungary
  • MIPT, Moscow, Russia
Gábor Tardos
  • Rényi Institute, Budapest, Hungary
  • MIPT, Moscow, Russia
Géza Tóth
  • Rényi Institute, Budapest, Hungary

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János Pach, Gábor Tardos, and Géza Tóth. Disjointness Graphs of Short Polygonal Chains. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 56:1-56:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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).

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph coloring
  • chi-bounded
  • disjointness graph


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