Mensa Polyhedral Graph Solutions

 

As noted in the original puzzle post (Oct. 28), a graph is designated Hamiltonian if there is a closed circuit along the edges of the graph that 'hits' each node exactly once. As seen above, a sketch of all the 2D forms for the Platonic solids shows all of the nodes are hit exactly once. Hence all the graphs are Hamiltonian.

A graph is Eulerian if there is a closed circuit that hits each edge exactly once. Only one such 2D graph meets that criterion, the octahedron, as shown in the above sketch.

 Meanwhile each of the 2D graphs above (L-R: tetrahedron, cube, octahedron) can be designated "graceful" given that for each:

A node gets a number from 0 to n, where n is the number of edges, and no node number is repeated.

Each edge has a number equal to the positive difference of the node numbers on either side of it. (e.g. for the cube, 9 - 7 = 2)

No edge number is repeated given edges are numbered 1 through n.