TITLE: What is the smallest uniquely hamiltonian graph with minimum degree at least 3?
QUESTION [15 upvotes]: I would like to know more about uniquely hamiltonian graphs with minimum vertex degree at least 3, and in particular what is the smallest one.
(Recall that a graph is hamiltonian if it has a cycle passing through each vertex exactly once, and is uniquely hamiltonian if there is only one such cycle.)
Here's the smallest one that I currently know.

Does anyone know if a smaller one (fewer vertices) has been published?

REPLY [9 votes]: The system encouraged me to answer my own question, although it feels a bit strange to do so.
Anyway, after a bit of thinking and a (more substantial) bit of computing, I can now safely conclude that this 18-vertex 28-edge graph is the smallest uniquely-hamiltonian graph with minimum degree 3, and there are no others of this order (number of vertices) and size (number of edges).