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 $$C_{n/2}$$ 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

In this paper, we disprove the long-standing conjecture that any complete geometric graph on 2n vertices can be partitioned into n plane spanning trees. Our construction is based on so-called bumpy wheel sets. We fully characterize which bumpy wheels can and in particular which cannot be partitioned into plane spanning trees (or even into arbitrary plane subgraphs). Furthermore, we show a sufficient condition for generalized wheels to not admit a partition into plane spanning trees, and give a complete characterization when they admit a partition into plane spanning double stars. Finally, we initiate the study of partitions into beyond planar subgraphs, namely into k-planar and k-quasi-planar subgraphs and obtain first bounds on the number of subgraphs required in this setting.