There are three of these cycles, passing through six of the eight degree-three vertices (red in the illustration).

[2] It has order-6 dihedral symmetry, for a total of 12 members of its automorphism group.

[4] This can be chosen so that each graph automorphism corresponds to a symmetry of the polyhedron, in which case three of the faces will be rhombi or squares, and the other six will be kites.

Polyhedral face numberings of this type are used as "spindown life counters" in the game Magic: The Gathering, to track player lives, by turning the polyhedron to an adjacent face whenever a life is lost.

A card in the game, the Lich, allows players to return from a nearly-lost state with a single life to their initial number of lives.

Thus, a cycle passing once through each of the eleven vertices cannot exist in the Herschel graph.

into two symmetric halves by three-vertex separators and then combining one half from each graph.

The "essentially 6-edge-connected" terminology means that this trivial way of disconnecting the graph is ignored, and it is impossible to disconnect the graph into two subgraphs that each have at least two vertices by removing five or fewer edges.

[13] The Herschel graph is named after Alexander Stewart Herschel, a British astronomer, who wrote an early paper concerning William Rowan Hamilton's icosian game.

This is a puzzle involving finding Hamiltonian cycles on a polyhedron, usually the regular dodecahedron.

The Herschel graph describes the smallest convex polyhedron that can be used in place of the dodecahedron to give a game that has no solution.

Herschel's paper described solutions for the Icosian game only on the graphs of the regular tetrahedron and regular icosahedron; it did not describe the Herschel graph.