Wagner, Andrew2013-12-052013-12-0520132013http://hdl.handle.net/10393/30290http://dx.doi.org/10.20381/ruor-3426In 1975, Sheehan conjectured that every simple 4-regular hamiltonian graph has a second Hamilton cycle. If Sheehan's Conjecture holds, then the result can be extended to all simple d-regular hamiltonian graphs with d at least 3. First, we survey some previous results which verify the existence of a second Hamilton cycle if d is large enough. We will then demonstrate some techniques for finding a second Hamilton cycle that will be used throughout this paper. Finally, we use these techniques and show that for certain 4-regular Hamiltonian graphs whose automorphism group is large enough, a second Hamilton cycle exists.enHamilton cycleregular graphSheehan's Conjectureautomorphism groupThesis