On the existence of reflection symmetric flexible realizations

Abstract

The existence of a flexible quasi-injective realization in the plane is characterized by the existence of a NAC-coloring, which is a surjective coloring of edges by red and blue such that every cycle is either monochromatic, or there are at least two red and at least two blue edges. The idea of NAC-colorings was adjusted to the rotation symmetric setting: there is a rotation symmetric flexible realization if and only if there is a NAC-coloring invariant under the rotation with a certain property. The existence of a reflection symmetric quasi-injective realization with a flex preserving the symmetry, which is the topic of this talk, is surprisingly more difficult. We introduce the concept of pseudo-RS-colorings: an edge coloring by red, blue and gold such that there is at least one blue and one red edge, changing all gold edges to red, resp. all to blue, yields NAC-colorings and blue and red interchange under the reflection. An almost red-blue cycle is a cycle that has exactly one gold edge. A pseudo-RS-coloring is an RS-coloring either if there is no almost red-blue cycle, or for every red-blue cycle, there is another pseudo-RS-coloring differing in a specific way on the cycle. Our main results are the following: if a graph admits a reflection symmetric flexible quasi-injective realization, then the graph has an RS-coloring. This necessary condition can be strengthened to exclude some RS-colorings that cannot come from a flex. On the other hand, we show that if a graph has an RS-coloring with no almost red-blue cycle, then it has a reflection symmetric flexible quasi-injective realization. There is also a construction of a reflection symmetric flex in a very special case of RS-colorings with an almost red-blue cycle, but the complete characterization is still open. This is joint work with Sean Dewar and Georg Grasegger.