I've been recently trying to understand the precise properties of Cayley graphs in order to inform my students about some group theory. From this answer I understand that if we consider Cayley directed graph with coloured edges (meaning $ G = (V, E) $, where $ E \subset V \times V \times C $ and $ C $ is a set of colours, [labels, essentially]), non-isomorphic groups can not have the same Cayley graphs. Answers to this post suggest that if we consider Cayley undirected uncoloured graph, then non-isomorphic groups absolutely can have the same Cayley graph (example would be $ C_{2n} $ and $ D_n $ with the set of two generators which are their own inverses). However, I am struggling to understand the other cases: if we consider
- Cayley directed uncoloured graphs
or (out of curiosity) - Cayley undirected coloured graphs
what the results would be? I see the accepted answer to the last post, but I can't quite understand it and the link is unfortunately broken. Also, at least in the first case, I think it's quite natural to require the generating set to be minimal (so, not $ 1, -1 $ for cyclic group; otherwise, what the point of direction would be).