The number of non-crossing perfect plane matchings is minimized (almost) only by point sets in convex position

It is well-known that the number of non-crossing perfect matchings of 2k points in convex position in the plane is Ck, the kth Catalan number. Garćıa, Noy, and Tejel proved in 2000 that for any set of 2k points in general position, the number of such matchings is at least Ck. We show that the equality holds only for sets of points in convex position, and for one exceptional configuration of 6 points. Introduction, notation, result Let S be a set of n = 2k points in general position (no three points lie on the same line) in the plane. Under a perfect matching of S we understand a geometric perfect matching of the points of S realized by k non-crossing segments. The number of perfect matchings of S will be denoted by pm(S). In general, pm(S) depends on (the order type of) S. Only for very special configurations an exact formula is known. The well-known case is that of points in convex position: Theorem 1 (Classic/Folklore/Everybody knows). If S is a set of 2k points in convex position, then pm(S) = Ck = 1 k+1 ( 2k k ) , the kth Catalan number. There are several results concerning the maximum and minimum possible values of pm(S) over all sets of size n. For the maximum possible value of pm(S), only asymptotic bounds are known. The best upper bound up to date is due to Sharir and Welzl [3] who proved that for any S of size n, we have pm(S) = O(10.05n). For the lower bound, Garćıa, Noy, and Tejel [2] constructed a family of examples which implies the bound of Ω(3nnO(1)); it was recently improved by Asinowski and Rote [1] to Ω(3.09n). As for the minimum possible value of pm(S) for sets of size n = 2k, Garćıa, Noy, and Tejel [2] showed that it is attained by sets in convex position, and thus, by Theorem 1, it is Ck = Ω(2 n/n3/2): Theorem 2 (Garćıa, Noy, and Tejel, 2000 [2]). For any set S of n = 2k points in general position in the plane, we have pm(S) ≥ Ck. However, to the best of our knowledge, the question of whether only sets in convex position have exactly Ck perfect matchings, was never studied. In this note we show that this is almost the case: there exists a unique (up to order type) exception shown in Figure 1: ∗Institut für Informatik, Freie Univesität Berlin. E-mail 〈[email protected]〉. Supported by the ESF EUROCORES programme EuroGIGA, CRP ‘ComPoSe’, Deutsche Forschungsgemeinschaft (DFG), grant FE 340/9-1. 1 ar X iv :1 50 2. 05 33 2v 1 [ cs .C G ] 1 8 Fe b 20 15 Figure 1: A set of six points in non-convex position that has five perfect matchings. Theorem 3. Let S be a planar set of 2k points in general position. We have pm(S) = Ck only if S is in convex position, or if k = 3 and S is a set with the order type as in Figure 1. We recall the recursive definition of Catalan numbers: C0 = 1; and for k ≥ 1,