Related to uniquely hamiltionian graphs.
For natural numbers $a,b$ define $(a,b)$ gadget $G$:
$G$ is finite simple graph. Two vertices $u,v$ are of degree $b$ and the rest of the vertices are of degree $a$. There is exactly one hamiltonian path $u-v$.
Q1: Does $(3,2)$ gadget exist?
Q2: Does $(4,2)$ gadget exist?
To get uniquely hamiltonian regular graph for Q2, take two copies of the gadget $G_1$ and $G_2$. Merge $u_1,u_2$ and $v_1,v_2$.
Likely the answer to Q1 is positive since there are uniquely hamiltonian graphs with minimum degree $3$ and maximum degree $4$, while the answer to Q2 is likely negative since it is conjectured that there are no uniquely hamiltonian $4$-regular graphs.
Both gadgets are regular graph with two edges subdivided once.