The Faulkner-Younger graphs (Faulkner and Younger 1974) are the cubic polyhedral nonhamiltonian graphs on 42 and 44 vertices illustrated above that are counterexamples to Tait's Hamiltonian graph conjecture. Like the Grinberg graphs on 42 and 44 vertices, the smaller graph can be constructed from the larger by deleting a suitable graph edge, removing its two endpoints, and joining the other two neighbors of each endpoint.
