A rolling polyhedron graph is a graph obtained by rolling a polyhedral solid along a board whose tiles match up with the faces of the polyhedron being rolled. The vertices of such a graph are are pairs consisting of bottom face index and orientation, and two vertices are connected by an edge if they are obtainable by rolling the solid from a board tile to an adjacent one.
The rolling tetrahedron graph is the cubical graph.
The rolling cube graph is a graph with 24 vertices (6 faces times 4 orientations) and 48 edges corresponding to rolling a cube on a subset
of a square board (i.e., a normal chessboard
or checkerboard). It turns out to be isomorphic
to the bipartite double graph of the cuboctahedral graph. When given a circular embedding,
the outer circle is a Hamiltonian cycle corresponding
to cube rolling around inside the perimeter of a 
rectangle 3 times, giving all 24 orientations of the
cube.
The rolling octahedron graph is the Nauru graph 
on 24 vertices and 36 edges.
The rolling icosahedron graph is graph on 120 vertices (20 vertices times 6 orientations) and 180 edges corresponding to rolling a regular
icosahedron on a regular triangular board. It is isomorphic
to the cubic symmetric graph 
.
Rolling polyhedron graphs can also be constructed for various other deltahedra.
