Flexibility and Movability in Cayley Graphs

Abstract

Let ${\varvec{\Gamma }} = (V,E)$ be a (non-trivial) finite graph with $\lambda : E \rightarrow {\mathbb {R}}_{+}$ an edge labeling of ${\varvec{\Gamma }}$. Let $\rho : V\rightarrow {\mathbb {R}}^{2}$ be a map which preserves the edge labeling, i.e.,

$$\begin{aligned} \Vert \rho (u) - \rho (v)\Vert _{2} = \lambda ((u,v)), \,\forall (u,v)\in E, \end{aligned}$$

where $\Vert x-y\Vert _{2}$ denotes the Euclidean distance between two points $x,y \in {\mathbb {R}}^{2}$. The labeled graph is said to be flexible if there exists an infinite number of such maps (up to equivalence by rigid transformations) and it is said to be movable if there exists an infinite number of injective maps (again up to equivalence by rigid transformations). We study movability of Cayley graphs and construct regular movable graphs of all degrees. Further, we give explicit constructions of dense, movable graphs.