Flexibility of braced Penrose tilings and other infinite frameworks

Jan Legerský, 14 Mar 2022

A framework, which is a graph with a realization of its vertices in the plane, is called flexible if it can be continuously deformed while preserving the distance between adjacent vertices. Recently, the existence of a flexible framework for a given finite graph was characterized by the existence of a special edge coloring, called a NAC-coloring.

In this talk, we extend this result to infinite graphs using König’s lemma. Moreover, we focus on a certain class of (countably infinite) frameworks with 4-cycles forming (braced) parallelograms. Such a framework is flexible if and only if the graph has a NAC-coloring satisfying an extra condition. Since the 1-skeleton of a Penrose tiling belongs to this class, its flexibility can be decided using the presented tools.