Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

Date and Time: Tuesday, October 06, 2015, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Emo Welzl

Crossing-Free Perfect Matchings in Wheel Configurations

Consider a planar finite point set P, no three on a line and exactly one point not extreme in P. We call this a wheel configuration 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:= |S|/2.)

We give exact tight upper and lower bounds on pm(P) dependent on |P|. Simplified to its asymptotics in terms of C_m, these yield

(9/8) C_m (1 - o(1)) $\le$ pm(P) $\le$ (3/2) C_m( 1- o(1))

We characterize the 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 configuration (this follows along a short excursion to well formed parentheses strings). Therefore, leaving convex position increases the number of crossing-free perfect matchings by at least a factor of 9/8 (in the limit as |S| grows). From what we show it follows that pm(P) can be computed efficiently.

A connection to related problems (triangulations, origin embracing triangles) is briefly discussed. (Joint work with Andres J. Ruiz-Vargas, EPFL.)

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