Perfect Matchings with Crossings

• Published in: Algorithmica Vol. 86; no. 3; pp. 697 - 716
• Main Authors: Aichholzer, Oswin, Fabila-Monroy, Ruy, Kindermann, Philipp, Parada, Irene, Paul, Rosna, Perz, Daniel, Schnider, Patrick, Vogtenhuber, Birgit
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.03.2024; Springer Nature B.V
• Subjects: Algorithm Analysis and Problem Complexity; Algorithms; Computer Science; Computer Systems Organization and Communication Networks; Data Structures and Information Theory; Matching; Mathematics of Computing; Theory of Computation; Order types; Combinatorial geometry; Crossings; Geometric graphs; Perfect matchings
• ISSN: 0178-4617; 1432-0541; 1432-0541
• DOI: 10.1007/s00453-023-01147-7

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.