Perfect k-colored matchings and (k+2)-gonal tilings

TL;DR

This paper establishes a simple bijection between geometric perfect matchings and polygonal tilings, extending to colored matchings and tilings, with implications for algebraic structures and efficient computation.

Contribution

It introduces a new bijection linking perfect matchings and polygonal tilings, generalizing to colored cases and providing linear-time computability.

Findings

Bijection between perfect matchings and triangulations

Extension to colored matchings and polygonal tilings

Linear-time computation of corresponding elements

Abstract

We derive a simple bijection between geometric plane perfect matchings on $2n$ points in convex position and triangulations on $n+2$ points in convex position. We then extend this bijection to monochromatic plane perfect matchings on periodically $k$-colored vertices and $(k+2)$-gonal tilings of convex point sets. These structures are related to a generalization of Temperley-Lieb algebras and our bijections provide explicit one-to-one relations between matchings and tilings. Moreover, for a given element of one class, the corresponding element of the other class can be computed in linear time.