Perfect Matchings with Crossings

Oswin Aichholzer, Ruy Fabila Monroy, Philipp Kindermann, Irene Parada, Rosna Paul, Daniel Perz, Patrick Schnider, Birgit Vogtenhuber

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 \(C_{n/2}\) 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 \(k\le \frac{1}{64}n^2-\frac{35}{32}n\sqrt{n}+\frac{1225}{64}n\), 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 \(\frac{5}{72}n^2-\frac{n}{4}\) 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 \(\left( {\begin{array}{c}n/2\\ 2\end{array}}\right) \) crossings and with \({\left( {\begin{array}{c}n/2\\ 2\end{array}}\right) }\!-\!1\) crossings.