The deltoidal hexecontahedral graph is an Archimedean dual graph which is the skeleton of the deltoidal hexecontahedron as well as the rhombic hexecontahedron. It is illustrated above in a couple of drawings.
It is implemented in the Wolfram Language as GraphData["DeltoidalHexecontahedralGraph"].
The plots above show the adjacency matrices, incidence matrices, and graph distance matrices for the deltoidal hexecontahedral graph.
While the above drawing contains overlapping edges, it still follows from this vertex coloring that
the deltoidal hexecontahedral graph is untraceable
and therefore also nonhamiltonian (T. León
and R. Winton, pers. comm., Jul. 15, 2006). This is true because the vertex degrees of adjacent
vertices in the solid alternate between even and
odd. Since there are 32 (
) combined degree-3 and
degree-5 vertices
but only 30 degree-4 vertices,
this imbalance rules out a Hamiltonian path.
The following table summarizes some properties of the graph.
