The snub icosidodecadodecahedral graph is a sextic Cayley graph on 60 nodes and 180 edges that corresponds to the skeleton of the snub icosidodecadodecahedron.
The snub icosidodecadodecahedral graph is Eulerian, Hamiltonian, and vertex-transitive.
It is implemented in the Wolfram Language as GraphData["SnubIcosidodecadodecahedralGraph"].