I don't know how one can tackle the following kind of question, so any hint is welcome. I formulate a precise question in order to fix ideas, but it is to be considered as an example out of a more general class.


**Example:** *Can one embed **[Petersen's graph](http://en.wikipedia.org/wiki/Petersen_graph)** in $\mathbb R^4$ in such a way that all edges are mapped to segments (of not necessarily equal lenght) and each pair of adjacent segments forms an angle of $2\pi/3$ ?*


The question can be generalized in the following way, to give a wider class of questions: 


* instead of Petersen's graph, consider a given $k$-regular graph (we had $k=3$ above)
* instead of $\mathbb R^4$ consider $\mathbb R^n, n>k$, or embeddings into the round sphere $S^n$ where the edges of the graph are mapped into geodesics
* ask that the angles formed between any pair of adjacent edges are equal to the angle $VOV'$ made at which two vertices $V,V'$ of a regular euclidean $(k-1)$-simplex are seen from its barycenter $O$


I think that $1$-skeletons of spherical regular polytopes give such kind of graphs.
I am interested in any partial answer which introduces essentially more ingredients than just constructions based on symmetry groups, or in obstructions which show negative answers.



When do there exist locally regular embeddings of regular graphs?
Assume $\Gamma$ is a 3-regular graph in $\mathbb{R}^n$ and the angles at each vertex are $\tfrac23\pi$.
Then it is easy to see that the distance to the origin can not have a local maximum on $\Gamma$. 


Therefore $\Gamma$ has to be infinite; in particular it can not be Petersen's graph.


