Point sets with many non-crossing matchings

Abstract

The maximum number of non-crossing straight-line perfect matchings that a set of $n$ points in the plane can have is known to be $O(10.0438^n)$ and ${\Omega }^{\ast }(3^n)$. The lower bound, due to García, Noy, and Tejel (2000) is attained by the double chain, which has $\Theta (3^n{n}^{O(1)})$ such matchings. We reprove this bound in a simplified way that uses the novel notion of down-free matching, and apply this approach on several other constructions. As a result, we improve the lower bound. First we show that double zigzag chain with $n$ points has ${\Theta }^{\ast }({\lambda }^n)$ such matchings with $\lambda \approx 3.0532$. Next we analyze further generalizations of double zigzag chains - double $r$-chains. The best choice of parameters leads to a construction with ${\Theta }^{\ast }({\nu }^n)$ matchings, with $\nu \approx 3.0930$. The derivation of this bound requires an analysis of a coupled dynamic-programming recursion between two infinite vectors.

33 pages, 19 figures, 2 tables

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