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 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.
|Title of host publication||Proc. $33^rd$ European Workshop on Computational Geometry EuroCG '17|
|Place of Publication||Malmö, Sweden|
|Number of pages||4|
|Publication status||Published - 2017|
Aichholzer, O., Andritsch, L., Baur, K., & Vogtenhuber, B. (2017). Perfect $k$-colored matchings and $k+2$-gonal tilings. In Proc. $33^rd$ European Workshop on Computational Geometry EuroCG '17 (pp. 81-84). Malmö, Sweden.