Point sets with many non-crossing perfect 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(10.0438n) and Ω⁎(3n). The lower bound, due to García, Noy, and Tejel (2000), is attained by the double chain, which has Θ(3n/nΘ(1)) such matchings. We reprove this bound in a simplified way that uses the novel notion of down-free matchings. We then apply this approach to several other constructions. As a result, we improve the lower bound. First we show that the double zigzag chain with n points has Θ⁎(λn) non-crossing perfect matchings with λ≈3.0532. Next we analyze further generalizations of double zigzag chains – double r-chains. The best choice of parameters leads to a construction that has Θ⁎(νn) matchings with ν≈3.0930. The derivation of this bound requires an analysis of a coupled dynamic-programming recursion between two infinite vectors.