The Robertson graph is the unique 
-cage graph, illustrated
above in a number of embeddings. It has 19 vertices and 38 edges. It has girth 5,
diameter 3, chromatic number 3, and is a quartic
graph. The third embedding above is due to Bondy and Murty (1976, p. 237).
The Robertson graph is implemented in the Wolfram Language as GraphData["RobertsonGraph"].
The Robertson graph has automorphism group order 24, possesses 5376 (directed) Hamiltonian cycles, and has 224 distinct order-1 generalized LCF notations, with none of higher order. Of these, 20 (illustrated above) have bilateral symmetry.
The Möbius-Kantor graph can be obtained as a subgraph of the Robertson graph by removing the three vertices and two edges illustrated above (pers. comm., E. Pegg, Jr., Oct. 27, 2025).
Computations show that the Robertson graph has graph crossing number and rectilinear crossing number at most 17. Two 17-crossing rectilinear embeddings are illustrated above. The first was found by G. Exoo, while the second was found by H. C. Silva Filho (pers. comm. to E. Pegg, Jr., Aug. 12, 2026).
The Robertson graph satisfies the rhombus constraints and contains no known unit-distance forbidden subgraph, yet appears not to be a unit-distance. A number of embeddings found from different initial embeddings by minimizing the sum of square deviations from unit edge lengths until a local minimum was reached are illustrated above.
