Robertson graph

In the mathematical field of graph theory, the Robertson graph or (4,5)-cage, is a 4-regular undirected graph with 19 vertices and 38 edges named after Neil Robertson.^[2][3]

The Robertson graph is the unique (4,5)-cage graph and was discovered by Robertson in 1964.^[4] As a cage graph, it is the smallest 4-regular graph with girth 5.

It has chromatic number 3, chromatic index 5, diameter 3, radius 3 and is both 4-vertex-connected and 4-edge-connected. It has book thickness 3 and queue number 2.^[5] The graph is neither planar nor 1-planar.^[6]

The Robertson graph is also a Hamiltonian graph which possesses 5,376 distinct directed Hamiltonian cycles.

The Robertson graph is one of the smallest graphs with cop number 4.^[7]

The Robertson graph is not a vertex-transitive graph; its full automorphism group is isomorphic to the dihedral group of order 24, the group of symmetries of a regular dodecagon, including both rotations and reflections.^[8]

The characteristic polynomial of the Robertson graph is

${\displaystyle (x-4)(x-1)^{2}(x^{2}-3)^{2}(x^{2}+x-5)}$

${\displaystyle (x^{2}+x-4)^{2}(x^{2}+x-3)^{2}(x^{2}+x-1).\ }$

• The Robertson graph as drawn in the original publication.
• The chromatic number of the Robertson graph is 3.
• The chromatic index of the Robertson graph is 5.