Asymmetric planar cubic graphs 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? **Background** 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).

Unless I'm mistaken, you can add a sufficiently large "cyclic ladder" to the Frucht graph as in: !alt text

I would argue that, in an automorphism, the right-hand side would need to be mapped to itself. Therefore, an automorphism of the above graph would imply an automorphism of the Frucht graph (and therefore must be trivial).

An infinite version could be constructed by appending two infinite ladders instead (actually, I'm pretty sure you could attach one infinite ladder).