For sets of n points, n even, in general position in the plane, we consider straight-line drawings of perfect matchings on them. It is well known that such sets admit at least different plane perfect matchings, where is the n/2-th Catalan number. Generalizing this result we are interested in the number of drawings of perfect matchings which have k crossings. We show the following results. (1) For every , any set with n points, n sufficiently large, admits a perfect matching with exactly k crossings. (2) There exist sets of n points where every perfect matching has at most crossings. (3) The number of perfect matchings with at most k crossings is superexponential in n if k is superlinear in n. (4) Point sets in convex position minimize the number of perfect matchings with at most k crossings for , and maximize the number of perfect matchings with crossings and with crossings.
Keywords: Combinatorial geometry; Crossings; Geometric graphs; Order types; Perfect matchings.
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