Hamiltonian cycles in symmetric graphs.

Let k be a positive integer. We define M[subscript]k to be the graph with a vertex set consisting of all binary strings of length 2k + 1 which have either k or k + 1 ones and edge set consisting of all pairs of these binary strings which differ in exactly one bit. Showing that the graph M[subscript]k is Hamiltonian for all k is known as the Middle Levels problem. This problem was first posed in the early 1980's and to this day remains unsolved. In this thesis we explore the symmetries of M[subscript]k and graphs related to it. We then use these symmetries to propose a method for finding Hamiltonian cycles in M[subscript]k when 2k + 1 and k are prime. We believe that our method is more efficient than methods proposed by previous authors.