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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.

enter image description here

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

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  • $\begingroup$ What are the numbers placed for? They definitely don't indicate the cycle (which is obvious). Do they bear some additional info? $\endgroup$
    – Ilya Bogdanov
    Nov 28, 2016 at 9:18
  • $\begingroup$ The numbers are just an arbitrary labelling so no further information there. $\endgroup$
    – Gordon Royle
    Nov 28, 2016 at 9:24
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    $\begingroup$ A google search brings up this article: link.springer.com/chapter/10.1007%2F978-3-319-39636-1_1. And it claims there is such a graph on only 10 vertices. $\endgroup$
    – Wolfgang
    Nov 28, 2016 at 11:10
  • $\begingroup$ @Wolfgang This is a very strange paper. I agree that it claims that there is such a graph on 10 vertices. However it does not actually contain any graphs, except for a diagram (Figure 1) of a 10-vertex graph that is not uniquely hamiltonian. The paper spends almost all its time on an extended description of a search algorithm that judging by the number of acronyms used must be quite complicated, and plenty of tables of running times and success rates etc. But no uniquely hamiltonian graphs :-( $\endgroup$
    – Gordon Royle
    Nov 28, 2016 at 12:05
  • $\begingroup$ Aha, figured it out. The 10-vertex graph in the aforementioned paper has the property that it has an edge that is contained in only one hamilton cycle. This means it can be turned into a uniquely hamiltonian graph via a simple doubling procedure. So while the 10-vertex graph itself is not uniquely hamiltonian, it leads to a 20-vertex example that is. $\endgroup$
    – Gordon Royle
    Nov 28, 2016 at 12:34

2 Answers 2

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I am not sure what the smallest such graph is, but since you also asked for more information on uniquely hamiltonian graphs with minimum degree $3$, Entringer and Swart proved the following nice theorem.

For each $n= 2k, k \geqslant 11$, there exists a uniquely hamiltonian graph on $n$ vertices having two vertices of degree $4$ and all others of degree $3$.

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  • $\begingroup$ Yes, I came across this paper, and wondered whether their 22-vertex example might be the smallest, which really started me off on trying to find the smallest. My guess is that my 18-vertex example is the smallest, but I haven't confirmed this yet. I do know that it is the only one with this number of vertices and edges. $\endgroup$
    – Gordon Royle
    Nov 28, 2016 at 10:14
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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).

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