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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.

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Data availability

Not applicable as the results presented in this manuscript rely on no external sources of data or code.

Notes

  1. A graph is said to be r-regular (where \(r\geqslant 1\) is an integer) if there are exactly r edges connected to each vertex.

  2. We use the convention that a finite group G is generated by a set S, if every element of G can be written as a product of elements in \(S\cup S^{-1}\).

  3. A slight modification is needed as we are discussing rigidity, flexibility and movability of loopless graphs whereas the square graph contains loops. When we pose this question, we mean the modified square graph with the loops removed.

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Acknowledgements

I wish to thank the anonymous reviewers for their constructive comments and suggestions which improved the article. I am grateful to Josef Schicho for a number of helpful discussions on rigidity and flexibility of graphs and for his encouragement in pursuing the work. The project was initiated while on a visit to the Johann Radon Institute for Computational and Applied Mathematics (RICAM) and the Johannes Kepler University (JKU), Linz. The author thanks the Fakultät für Mathematik, Universität Wien where his work was supported by the European Research Council (ERC) grant of Goulnara Arzhantseva, “ANALYTIC” grant agreement no. 259527.

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Communicated by Torsten Ueckerdt.

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Biswas, A. Flexibility and Movability in Cayley Graphs. Ann. Comb. 26, 205–220 (2022). https://doi.org/10.1007/s00026-022-00569-4

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Keywords

  • Combinatorial rigidity
  • Cayley graphs
  • rigidity and flexibility of Cayley graphs
  • graph products

Mathematics Subject Classification

  • 52C25
  • 70B15
  • 05C15
  • 05C25
  • 05C38
  • 51K99