Disjoint Edges in Topological Graphs and the Tangled-Thrackle Conjecture
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.
KeywordsEdge Incident Intersection Graph Absolute Constant Topological Graph Horizontal Strip
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