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...

1 Answer

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.)

• How many of them are bipartite? – draks ... Jan 19 '15 at 6:23
• Here they say that $5$ of them are bipartite. Can you confirm that? – draks ... Jan 21 '15 at 17:11
• oh and here they are shown...only "$12$-cubic graph 66" looks asymmetric to me! – draks ... Jan 21 '15 at 17:14