Constructing all non-isomorphic Hamiltonian graphs on n vertices

Question

I've recently been taking a combinatorics class and have become interested in Hamiltonian graphs.

This OEIS entry lists the number of Hamiltonian graphs on $n$ vertices up to $n =12$. However, I am having difficulty finding how these numbers were found. Is it naive to expect an algorithm which actually constructs all the hamiltonian graphs (at least for $n$ not too large) to exist?

Thanks.

Answer 1

I expect by constructing all graphs, probably using Brendan McKay’s program “geng” and extracting the Hamiltonian ones. Needs quite a few computer cores to do n=12, and I wouldn’t try 13.

Answer 2

The 'filter' approach of generating all non-isomorphic graphs on $n$ vertices is the simplest possibility.

The OEIS entry references McKay (1996), but I can't see a paper in this list : https://users.cecs.anu.edu.au/~bdm/publications.html for Hamiltonian graphs (only _hypo_hamiltonian graphs).

Presumably a more efficient approach than filtering would be to extend a Hamiltonian graph on $n$ vertices to one on $n + 1$ vertices, but maybe that is not possible.