"According to a strengthened version of Frucht's theorem, there are infinitely many asymmetric cubic graphs." If there are any, what is the smallest cubic bipartite asymmetric graph?
Kind of a bipartite version of Frucht's graph. If there are none, why's that?
EDIT: The graph doesn't necessarily need to be planar, but $3$-edge-colorable...
There are 85 cubic graphs on 12 vertices and five of them are asymmetric, and all five are 3-edge colourable. No cubic graph on 10 vertices is asymmetric; I did not bother to check the graphs on eight vertices, because my recollection was that the smallest asymmetric regular graphs are on 12 vertices. (All computations in sage.)