only trivial automorphism on Frucht graph

Question

Why is there only the trivial automorphism on the Frucht graph?

We have a rooted tree in the Frucht graph which allows to totally order the vertices. But how does this imply that there is only the identity on the Frucht graph?

Answer 1

The graph contains exactly three triangles (induced graphs isomorphic to $K_3$) so these must be permuted by any automorphism. Now look at the distances between these triangles: an automorphism must preserve them; exactly one triangle is one edge away from the other two, so it must be preserved by an isomorphism. This almost gets you there. Now finish :-)

Alternatively, there are three vertices which are not in triangles, so they must be permuted among themselves by an automorphism. Exactly one of these three vertices, call it $v$, is at distance three from one of the triangles, so it must be fixed by any automorphism; now exacty one of the other special vertices is at distance one from $v$, so it must be fixed, and the other special vertex must then also be fixed. We thus see that the three vertices not on triangles must be fixed. Continue in this way.