# Graph gadget related to uniquely hamiltionian regular graphs (question #2)

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.

Surely the answer to Q1 is negative. If you found such a graph, then by joining two copies together (joining $u$ in one copy to $u$ in the second, and the same for $v$), would this not give a 3-regular uniquely hamiltonian graph?