Up to a double cover, every regular connected graph is isomorphic to a Schreier graph
Paul-Henry Leemann
Abstract
We prove that every connected locally finite regular graph has a double cover which is isomorphic to a Schreier graph.
Table of Contents
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Up to a double cover, every regular connected graph is isomorphic to a Schreier graph
Authors
Abstract
We prove that every connected locally finite regular graph has a double cover which is isomorphic to a Schreier graph.
Paper Structure
This paper contains 4 sections, 8 theorems, 2 figures.
Table of Contents
Key Result
Proposition 1
Let $G$ be a $d$- regular connected graph. Then either $G$ is isomorphic to a Schreier graph or $G$ has a double-cover $H$ which is isomorphic to a Schreier graph.
Figures (2)
- Figure 1: Double covers of some graphs; the middle and the right ones are canonical double covers. The leftmost one is isomorphic to the double cover $\mathop{\mathrm{Cayl}}\nolimits(\mathbf{Z}/4\mathbf{Z};\{\pm1\})\twoheadrightarrow\mathop{\mathrm{Schrei}}\nolimits(\mathbf{Z}/4\mathbf{Z},\mathbf{Z}/2\mathbf{Z};\{\pm1\})$, the middle one to $\mathop{\mathrm{Schrei}}\nolimits(\mathbf{Z}/4\mathbf{Z},\mathbf{Z}/2\mathbf{Z};\{\pm1\})\twoheadrightarrow\mathop{\mathrm{Schrei}}\nolimits(\mathbf{Z}/4\mathbf{Z},\mathbf{Z}/4\mathbf{Z};\{\pm1\})$ and the rightmost one to $\mathop{\mathrm{Cayl}}\nolimits(\mathbf{Z}/2\mathbf{Z};\{\pm1\})\twoheadrightarrow\mathop{\mathrm{Schrei}}\nolimits(\mathbf{Z}/2\mathbf{Z},\mathbf{Z}/2\mathbf{Z};\{\pm1\})$.
- Figure 2: A simple $3$-regular graph (on the left) and a $4$-regular graph (on the right); both of them being not isomorphic to a Schreier graph.
Theorems & Definitions (15)
- Proposition 1
- Definition 2
- Remark 3
- Definition 4
- Definition 5
- Lemma 6
- proof
- Corollary 7
- Lemma 8
- Proposition 9: LeemannThese
- ...and 5 more