 The "middle-level" subgraph formula_83 is vertex-transitive (that is, its automorphism group is transitive, so that each vertex has the same "local environment"" and cannot be differentiated from the others, since we can relabel the coordinates as well as the binary digits to obtain an automorphism) and the problem of finding a Hamiltonian path in this subgraph is called the "middle-levels problem", which can provide insights into the more general conjecture.The question has been answered affirmatively for formula_84, and the preceding construction for monotonic codes ensures a Hamiltonian path of length at least 0.839"N" where "N" is the number of vertices in the middle-level subgraph.Another type of Gray code, the Beckett–Gray code, is named for Irish playwright Samuel Beckett, who was interested in symmetry.