Up to a double cover, every regular connected graph is isomorphic to a Schreier graph

We prove that every connected locally finite regular graph has a double cover which is isomorphic to a Schreier graph. The aim of this short note is to provide a proof of the following 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. While we were not able to find a reference to the above result in the literature, we do not claim any priority on it. In fact, this note was inspired by the following remark, which can be found at the end of Section 7 of [4]: “In fact up to the cover of degree 2 any regular graph can be realized as a Schreier graph [8]”. However, [8] seems to treat only the case of graphs of even degree. Section 1 contains all the definitions as well as a discussion on the unusual concept of degenerated loop. The short Section 2 contains more materials on coverings and perfect matchings, as well as the proof of Proposition 1. Acknowledgements The author is thankful to A. Georgakopoulos, R. Grigorchuk and M. de la Salle for comments on a previous version of this note. The author was supported by NSF Grant No. 200021_188578.