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 embeddings.
It is implemented in the Wolfram Language as GraphData["DeltoidalHexecontahedralGraph"].
The plots above show the adjacency, incidence, and graph distance matrices for the deltoidal hexecontahedral graph.
While the above embedding contains overlapping edges, it still follows from this 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, there will
always be one odd vertex than cannot be reached by any path.
The following table summarizes some properties of the graph.
| property | value |
| automorphism group order | 120 |
| characteristic polynomial |  |
| chromatic number | 2 |
| claw-free | no |
| clique number | 2 |
| determined by spectrum | ? |
| diameter | 8 |
| distance-regular graph | no |
| dual graph name | small rhombicosidodecahedral graph |
| edge chromatic number | 5 |
| edge connectivity | 3 |
| edge count | 120 |
| Eulerian | no |
| girth | 4 |
| Hamiltonian | no |
| Hamiltonian cycle count | 0 |
| Hamiltonian path count | 0 |
| integral graph | no |
| independence number | 32 |
| line graph | ? |
| line graph name | 20-cycle graph |
| perfect matching graph | no |
| planar | yes |
| polyhedral graph | yes |
| polyhedron embedding names | deltoidal hexecontahedron |
| radius | 6 |
| regular | no |
| square-free | no |
| traceable | no |
| triangle-free | yes |
| vertex connectivity | 3 |
| vertex count | 62 |
