Sheehan's conjecture on uniquely Hamiltonian regular graphs

A finite graph is $r$-regular if every vertex has degree $r$, and a Hamilton cycle is a cycle containing every vertex of the graph exactly once.

Sheehan's conjecture. There is no finite $r$-regular graph with a unique Hamilton cycle for any $r>2$.

This conjecture concerns whether uniqueness of a Hamilton cycle can occur in finite regular graphs of degree greater than two. It is still open, although partial results have been proved.