Let a symmetrically $3d$ embeddable graph be a graph that can be embedded into $3d$ so that the embedding is arc-transitive, which means that every vertex-edge pair with the vertex incident to the edge can be mapped to any other vertex-edge pair using a symmetry of $3d$ space in such a way that every vertex maps to a vertex and every edge maps to an edge. What is the highest girth such a finite cubic graph can have? What is the highest girth such a cubic infinite graph can have, with the additional constraint that there must be a bounded distance between vertices?
The dodecahedron has girth $5$, and that is the highest girth that I know of for such a finite graph. The Laves graph is infinite and has a girth of $10$. Is an infinite tree possible? What if the graphs don't need to be cubic but have a degree greater than $2$? What if the minimum distance between vertices has to be the length of an edge?