Perfect k-Colored Matchings and ( k + 2 ) -Gonal Tilings

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.