# Graph gadget related to uniquely hamiltionian regular graphs

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.

The $4$-regular graph is two copies of $G$: $G_1$ and $G_2$ and additional edges $u_1 u_2$ and $v_1 v_2$.
• @domotorp There is no such conjecture about hamilton cycles through an edge in a 4-regular graph, as hamiltonian 4-regular graphs appear to have plenty of hamilton cycles, not just "a second", and the data for small graphs shows that pretty much anything can happen. In fact, if an edge lies in one hamilton cycle, then it seems to lie in at least six (and this only happens for $K_5$) – Gordon Royle Feb 3 '17 at 14:13