In [OEIS A003436](https://oeis.org/A003436), it is written that the number of inequivalent labeled Hamilton Cycles of an n-dimesnional [Octahedron](https://en.wikipedia.org/wiki/Octahedron) is the same as the number of Perfect Matchings in a the complement of the [Cycle Graph](https://en.wikipedia.org/wiki/Cycle_graph) on 2n vertices (i.e. a complete graph $K\_{2n}$ without the cycle $C\_{2n}$).


I can verify this relationship, but it seems very arbitrary to me. Unfortunatly, OEIS does not give further details.



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> Why are these number of these two properties (inequivalent Hamiltonian Cycles and Perfect Matchings, respectively) equivalent in these two different objects (n-dimensional Octahedron and Complement of Cycle Graph $C\_{2n}$)?
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And hint or reference to the literature is very much apprechiated.



Why is the number of Hamiltonian Cycles of n-octahedron equivalent to the number of Perfect Matching in specific family of Graphs?
The $n$-dimensional analogue of the octahedron is the complement of a perfect matching of its vertex set. (Every vertex is joined to every other vertex except its antipode.) If you take a Hamilton cycle in the $n$-dimensional octahedron then you can think of that as a labelling of the $2n$-cycle, and the non-edges of the $n$-octahedron give a perfect matching of the complement of this graph. 


This paragraph was added in answer to Nicodean's comment below. Label the vertices of the $n$-dimensional octahedron $1^+,1^-,...,n^+,n^-$, where the vertices $i^+$ and $i^-$ are antipodes. A labelled Hamilton cycle in the octahedral graph is then exactly a $2n$-cycle made from the $2n$ symbols $i^+,i^-$ with the property that the symbols $i^+$ and $i^-$ are never adjacent. Giving a perfect matching on the complement of a $2n$-cycle is the same as giving a matching of the elements of a $2n$-cycle so that adjacent elements are never matched. To do this, I choose the symbols making up my matching to be the symbols $1^+,1^-,...,n^+,n^-$, where the symbols $i^+$ and $i^-$ are pairs that are matched. The condition that adjacent symbols are never matched is exactly the same condition that I had on the $2n$-cycles that describe the Hamilton cycles in the $n$-octahedral graph. 


One reason why this seems hard is that the matching sort of stays the same, while the $2n$-cycle whose complement it lives in is the thing that changes when you change the Hamilton cycle. 


