Perfect Matchings with Crossings - Aichholzer, Oswin; Fabila-Monroy, Ruy; Kindermann, Philipp; Parada, Irene; Paul, Rosna; Perz, Daniel; Schnider, Patrick; Vogtenhuber, Birgit (Algorithmica, 2024)
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 C n / 2 different plane perfect matchings, where C n / 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 k ≤ 1 64 n 2 - 35 32 n n + 1225 64 n , 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 72 n 2 - n 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 n / 2 2 crossings and with n / 2 2 - 1 crossings.