Hamiltonian circuit on a small rhombicosidodecahedron

Although every archimedean polyhedron has a hamiltonian circuit between its vertices,

I was not able to find a general algorithm to find one of them. Just used a enumeration algorithm for the main part of the path, then finishing by hand.

Any pointer to the proof of sentence 1, or to an easier algorithm will be much appreciated.

Both preceeding pictures are more mathematically correct--->

Done