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Crossing numbers Pach, János


One of the most useful tools in topological graph theory is the so-called Crossing Lemma of Ajtai, Chvatal, Newborn, Szemeredi (1982) and Leighton (1983). It states, roughly speaking, that if a graph drawn in the plane has much more edges than vertices, then the number of crossings between its edges is much larger than the number of edges. We extend this result to simple topological multigraphs, that is, for multigraphs drawn in the plane such that (1) any two independent edges meet in at most one point, (2) no two edges that share an endpoint have any interior point in common, and (3) both lenses enclosed by two edges that have the same endpoint contain at least one vertex in their interiors. Joint work with Geza Toth.

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