# Crossing-Free Perfect Matchings in Wheel Point Sets

Chapter

## Abstract

Consider a planar finite point set P, no three points on a line and exactly one point not extreme in P. We call this a wheel set and we are interested in pm(P), the number of crossing-free perfect matchings on P. (If, contrary to our assumption, all points in a set S are extreme, i.e. in convex position, then it is well-known that pm(S) = C m , the mth Catalan number, $$m:= \frac{\vert S\vert } {2}$$.)

We give exact tight upper and lower bounds on pm(P) depending on the cardinality of the wheel set P. Simplified to its asymptotics in terms of C m , these yield
$$\displaystyle{ \frac{9} {8}C_{m}(1 + o(1)) \leq \mathsf{pm}(P) \leq \frac{3} {2}C_{m}(1 + o(1))\,m:= \frac{\vert P\vert } {2}. }$$
We characterize the wheel sets (order types) which maximize or minimize pm(P). Moreover, among all sets S of a given size not in convex position, pm(S) is minimized for some wheel set. Therefore, leaving convex position increases the number of crossing-free perfect matchings by at least a factor of $$\frac{9} {8}$$ (in the limit as | S | grows). We can also show that pm(P) can be computed efficiently.

A connection to origin embracing triangles is briefly discussed.

## Notes

### Acknowledgements

We thank Vera Rosta, Patrick Schnider, Shakhar Smorodinsky, Antonis Thomas, and Manuel Wettstein for discussions on and suggestions for the material covered in this paper. We also thank the three referees for carefully reading through the paper with numerous suggestions for improvements. (And thanks to Maple, for sparing us some tedious calculations.)

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