At Wikipedia I found that "according to a strengthened version of Frucht's theorem, there are infinitely many asymmetric cubic graphs".
Are there infinitely many asymmetric planar cubic graphs, too?
If so, does it follow that there is an infinite asymmetric planar cubic graph?
If so, how could this graph be characterized?
I am looking for an homogeneous and isotropic (in the large) regular graph that could "mimick" a discretized plane (without distinguished directions as in a grid). So an infinite asymmetric 4-regular graph would be even better (reflecting dimension 2).