Perfect Matchings with Crossings

Abstract

AbstractFor 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 $$C_{n/2}$$ C n / 2 different plane perfect matchings, where $$C_{n/2}$$ 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 $$k\le \frac{1}{64}n^2-\frac{35}{32}n\sqrt{n}+\frac{1225}{64}n$$ 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 $$\frac{5}{72}n^2-\frac{n}{4}$$ 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 $$k=0,1,2$$ k = 0 , 1 , 2 , and maximize the number of perfect matchings with $$\left( {\begin{array}{c}n/2\\ 2\end{array}}\right) $$ n / 2 2 crossings and with $${\left( {\begin{array}{c}n/2\\ 2\end{array}}\right) }\!-\!1$$ n / 2 2 - 1 crossings.