Abstract
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 Cn/2 different plane perfect matchings, where Cn/2 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 [Formula: see text] , 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 5/72n2 - 4/ 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 k = 0, 1, 2, and maximize the number of perfect matchings with [Formula: see text] crossings and with [Formula: see text] crossings.