The rhombic dodecahedral graph is the Archimedean dual graph which is the skeleton of the rhombic dodecahedron (as well as the Bilinski dodecahedron). It is the Levi graph of the Miquel configuration. The rhombic dodecahedral graph is bipartite, edge-transitive, nonhamiltonian, planar, polyhedral, and untraceable. It is illustrated above in a number of drawings.
The graph was rediscovered by A. Fruchard (Maddaloni and Zamfirescu 2016) for its property of being small (14 vertices), polyhedral, and untraceable. For this reason, it was termed the "Fruchard graph" by Maddaloni and Zamfirescu (2016) and van Cleemput and Zamfirescu (2018), apparently without realizing its origin as the skeleton of the rhombic dodecahedron. The rhombic dodecahedral graph is however not alone in having these properties; the small triakis octahedral graph is another 14-vertex polyhedral untraceable graph.
The rhombic dodecahedral graph is also its own distance-3 graph distance graph.
The rhombic dodecahedral graph is implemented in the Wolfram Language as GraphData["RhombicDodecahedralGraph"].
The plots above show the adjacency matrix, incidence matrix, and graph distance matrix for the rhombic dodecahedral graph.
The following table summarizes some properties of the graph.
| property | value |
| automorphism group order | 48 |
| characteristic polynomial |  |
| chromatic number | 2 |
| chromatic polynomial |  |
| claw-free | no |
| clique number | 2 |
| determined by spectrum | ? |
| diameter | 4 |
| distance-regular graph | no |
| dual graph name | cuboctahedral graph |
| edge chromatic number | 4 |
| edge connectivity | 3 |
| edge count | 24 |
| Eulerian | no |
| girth | 4 |
| Hamiltonian | no |
| Hamiltonian cycle count | 0 |
| Hamiltonian path count | 0 |
| integral graph | no |
| independence number | 8 |
| line graph | ? |
| perfect matching graph | no |
| planar | yes |
| polyhedral graph | yes |
| polyhedron embedding names | rhombic dodecahedron |
| radius | 4 |
| regular | no |
| square-free | no |
| traceable | no |
| triangle-free | yes |
| vertex connectivity | 3 |
| vertex count | 14 |
