Abstract
It is shown that for a constant t ∈ ℕ, every simple topological graph on n vertices has O(n) edges if the graph has no two sets of t edges such that every edge in one set is disjoint from all edges of the other set (i.e., the complement of the intersection graph of the edges is K t,t -free). As an application, we settle the tangled-thrackle conjecture formulated by Pach, Radoičić, and Tóth: Every n-vertex graph drawn in the plane such that every pair of edges have precisely one point in common, where this point is either a common endpoint, a crossing, or a point of tangency, has at most O(n) edges.
Work on this paper began at the AIM workshop Exact Crossing Numbers (Palo Alto, CA, 2014). Research by Ruiz-Vargas was supported by the Swiss National Science Foundation grants 200021-125287/1 and 200021-137574. Research by Suk was supported by an NSF Postdoctoral Fellowship and by the Swiss National Science Foundation grant 200021-125287/1. Research by Tóth was supported in part by the NSF award CCF 1423615.
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Ruiz-Vargas, A.J., Suk, A., Tóth, C.D. (2014). Disjoint Edges in Topological Graphs and the Tangled-Thrackle Conjecture. In: Duncan, C., Symvonis, A. (eds) Graph Drawing. GD 2014. Lecture Notes in Computer Science, vol 8871. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-45803-7_24
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DOI: https://doi.org/10.1007/978-3-662-45803-7_24
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