What is the smallest planar cubic bipartite asymmetric graph? "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...
 A: 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.)
A: Using a combination of the geng and IGraph/M tools, I find none with less than 18 vertices. With 18 vertices, there are 13 of them:
Mathematica code:
Needs["IGraphM`"]
grs = Select[
  IGImport["!geng -d3 -D3 -b 18", "Nauty"],
  IGBlissAutomorphismGroup[#] === PermutationGroup[{}] &
]


In graph6 notation (so that you can compute with them e.g. in Sage),
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The smallest ones that are also planar have 22 vertices:
grs = Select[IGImport["!/opt/local/bin/geng -d3 -D3 -b 22", "Nauty"], 
   IGBlissAutomorphismGroup[#] === PermutationGroup[{}] &];

Select[grs, IGPlanarQ]


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