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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?

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?

Added years later (2023)

With almost any extremal question in Hamiltonicity the Petersen graph crops up and this is no exception.

This picture shows that the graph can be partitioned into two 9-vertex induced subgraphs, each obtained by deleting a vertex from the Petersen graph and then joined by four judiciously chosen edges.

The graph can be shown to be uniquely Hamiltonian by arguing in the "half-graph" about how paths can connect the four "terminals" (vertices with edges to the other side).

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 each, 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?

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?