[2] It is named after Harold Scott MacDonald Coxeter.

[3] The Coxeter graph is hypohamiltonian: it does not itself have a Hamiltonian cycle but every graph formed by removing a single vertex from it is Hamiltonian.

It has rectilinear crossing number 11, and is the smallest cubic graph with that crossing number[4] (sequence A110507 in the OEIS).

The simplest construction of a Coxeter graph is from a Fano plane.

There is an independent set of size 15 that includes v. Delete the 7 neighbors of v, and the whole independent set including v, leaving behind the Coxeter graph.

[6] It acts transitively on the vertices, on the edges and on the arcs of the graph.

[8] As a finite connected vertex-transitive graph that contains no Hamiltonian cycle, the Coxeter graph is a counterexample to a variant of the Lovász conjecture, but the canonical formulation of the conjecture asks for an Hamiltonian path and is verified by the Coxeter graph.

These are different representations of the Coxeter graph, using the same vertex labels.

Each red, green or blue vertex is connected with two vertices of the same color (thin edges forming 7-cycles) and to one white vertex (thick edges).