Here are some usefull facts, and some historical details.

Every $2d$-regular graph (without loops of degree 1) is isomorphic to a Schreier graph.

The results for finite graphs is due to Gross. The result for locally finite graphs follows by compacity and was probably one of this well-known "folklore result" for long. To my knowledge, the first written proof of it can be found in "Some problems of the dynamics of group actions on rooted trees" by R. Grigorchuk 2011. But it has since be "reproven" many times (by J. Cannizzo in 2013, in my own phd thesis around the same time, ...).

A (2d+1)-regular graph (without loops of degree 1) is isomorphic to a Schreier graph if and only if it has a perfect matching.

This follows from the previous fact. See for example ARG's answer.

Every countable vertex transitive graphs is isomorphic to a Schreier graphs.

The finite case was done by C. Godsil and G. Royle in 2001. I did (circa 2013) the locally finite case in my phd thesis, using results of Aharoni on matchings in infinite graphs. I believe that an independant proof of it was published in 2016 by another author, but sadly cannot remember more details. Finally, the countable case was done by M. Hamman and A. Wendland in the appendix of https://arxiv.org/abs/2007.06432

Up to a double cover, any locally finite regular graphs is isomorphic to a Schreier graph.

See https://arxiv.org/pdf/2010.06431.pdf, which is intended as a small note for reference.

Here is the skeleton of the proof. Let $G$ be a locally finite graph. If $G$ is bipartite, put $K=G$. Otherwise, put $K=G\times C_2$. Then $K$ is connected, bipartite (and hence without loops of degree 1), regular and covers $G$. Since $K$ is bipartite and regular it has a perfect matching. The desired result follows.

Here are some usefull facts, and some historical details.

Every $2d$-regular graph (without loops of degree 1) is isomorphic to a Schreier graph.

The results for finite graphs is due to Gross. The result for locally finite graphs follows by compacity and was probably one of this well-known "folklore result" for long. To my knowledge, the first written proof of it can be found in "Some problems of the dynamics of group actions on rooted trees" by R. Grigorchuk 2011. But it has since be "reproven" many times (by J. Cannizzo in 2013, in my own phd thesis around the same time, ...).

A (2d+1)-regular graph (without loops of degree 1) is isomorphic to a Schreier graph if and only if it has a perfect matching.

This follows from the previous fact. See for example ARG's answer.

Every countable vertex transitive graphs is isomorphic to a Schreier graphs.

The finite case was done by C. Godsil and G. Royle in 2001. I did (circa 2013) the locally finite case in my phd thesis, using results of Aharoni on matchings in infinite graphs. An independant proof of it was published by Csoka and Lippner in 2017 https://ems.press/journals/ggd/articles/14727. Finally, the countable case was done by M. Hamman and A. Wendland in the appendix of https://arxiv.org/abs/2007.06432

Up to a double cover, any locally finite regular graphs is isomorphic to a Schreier graph.

See https://arxiv.org/pdf/2010.06431.pdf, which is intended as a small note for reference.

Here is the skeleton of the proof. Let $G$ be a locally finite graph. If $G$ is bipartite, put $K=G$. Otherwise, put $K=G\times C_2$. Then $K$ is connected, bipartite (and hence without loops of degree 1), regular and covers $G$. Since $K$ is bipartite and regular it has a perfect matching. The desired result follows.

Here are some usefull facts, and some historical details.

Every $2d$-regular graph (without loops of degree 1) is isomorphic to a Schreier graph.

The results for finite graphs is due to Gross. The result for locally finite graphs follows by compacity and was probably one of this well-known "folklore result" for long. To my knowledge, the first written proof of it can be found in "Some problems of the dynamics of group actions on rooted trees" by R. Grigorchuk 2011. But it has since be "reproven" many times (by J. Cannizzo in 2013, in my own phd thesis around the same time, ...).

A (2d+1)-regular graph (without loops of degree 1) is isomorphic to a Schreier graph if and only if it has a perfect matching.

This follows from the previous fact. See for example ARG's answer.

Every countable vertex transitive graphs is isomorphic to a Schreier graphs.

The finite case was done by C. Godsil and G. Royle in 2001. I did (circa 2013) the locally finite case in my phd thesis, using results of Aharoni on matchings in infinite graphs. I believe that an independant proof of it was published in 2016 by another author, but sadly cannot remember more details. Finally, the countable case was done by M. Hamman and A. Wendland in the appendix of https://arxiv.org/abs/2007.06432

Up to a double cover, any locally finite graphs is isomorphic to a Schreier graph.

See https://arxiv.org/pdf/2010.06431.pdf, which is intended as a small note for reference.

Here is the skeleton of the proof. Let $G$ be a locally finite graph. If $G$ is bipartite, put $K=G$. Otherwise, put $K=G\times C_2$. Then $K$ is connected, bipartite (and hence without loops of degree 1), regular and covers $G$. Since $K$ is bipartite and regular it has a perfect matching. The desired result follows.

Here are some usefull facts, and some historical details.

Every $2d$-regular graph (without loops of degree 1) is isomorphic to a Schreier graph.

The results for finite graphs is due to Gross. The result for locally finite graphs follows by compacity and was probably one of this well-known "folklore result" for long. To my knowledge, the first written proof of it can be found in "Some problems of the dynamics of group actions on rooted trees" by R. Grigorchuk 2011. But it has since be "reproven" many times (by J. Cannizzo in 2013, in my own phd thesis around the same time, ...).

A (2d+1)-regular graph (without loops of degree 1) is isomorphic to a Schreier graph if and only if it has a perfect matching.

This follows from the previous fact. See for example ARG's answer.

Every countable vertex transitive graphs is isomorphic to a Schreier graphs.

The finite case was done by C. Godsil and G. Royle in 2001. I did (circa 2013) the locally finite case in my phd thesis, using results of Aharoni on matchings in infinite graphs. I believe that an independant proof of it was published in 2016 by another author, but sadly cannot remember more details. Finally, the countable case was done by M. Hamman and A. Wendland in the appendix of https://arxiv.org/abs/2007.06432

Up to a double cover, any locally finite regular graphs is isomorphic to a Schreier graph.

See https://arxiv.org/pdf/2010.06431.pdf, which is intended as a small note for reference.

Here is the skeleton of the proof. Let $G$ be a locally finite graph. If $G$ is bipartite, put $K=G$. Otherwise, put $K=G\times C_2$. Then $K$ is connected, bipartite (and hence without loops of degree 1), regular and covers $G$. Since $K$ is bipartite and regular it has a perfect matching. The desired result follows.