The ditrigonal icosidodecahedral graph is the skeleton of the cube 5-compound, ditrigonal dodecadodecahedron, great ditrigonalIcosidodecahedron, and small ditrigonal icosidodecahedron.
It is Hamiltonian with 
Hamiltonian cycles.
It is implemented in the Wolfram Language as GraphData["DitrigonalIcosidodecahedralGraph"].
