Abstract
A topological graph G is a graph drawn in the plane so that its edges are represented by Jordan arcs. G is called simple, if any two edges have at most one point in common. It is shown that the maximum number of edges of a simple topological graph with n vertices and no k pairwise disjoint edges is O(nlog4k − 8 n) edges. The assumption that G is simple cannot be dropped: for every n, there exists a complete topological graph of n vertices, whose any two edges cross at most twice.
János Pach has been supported by NSF Grant CCR-00-98246, by PSC-CUNY Research Award 65392-0034, OTKA T-030012, and by OTKA T-032452. Géza Tóth has been supported by OTKA T-030012 and OTKA T-038397.
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Pach, J., Tóth, G. (2005). Disjoint Edges in Topological Graphs. In: Akiyama, J., Baskoro, E.T., Kano, M. (eds) Combinatorial Geometry and Graph Theory. IJCCGGT 2003. Lecture Notes in Computer Science, vol 3330. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-30540-8_15
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