A graph is uniquely hamiltonian if it has exactly one hamiltonian cycle.
According to a conjecture there are no $r$-regular uniquely hamiltonian graphs for $r > 2$ and of special interest is the case $r=4$ ($r=3$ is solved).
The following graph gadget (if it exists) will give $4$-regular uniquely hamiltonian graph:
$G$ is finite simple connected graph. Two vertices $u,v$ are of degree $3$ and the rest vertices are of degree $4$. There is exactly one $u-v$ hamiltonian path.
Does such gadget exist?
Computer search didn't find any on up to 12 vertices.
The $4$-regular graph is two copies of $G$: $G_1$ and $G_2$ and additional edges $u_1 u_2$ and $v_1 v_2$.