# Reconstructing the number of Hamiltonian cycles

As is common terminology in graph reconstruction, given a graph $G$, we call a vertex deleted subgraph of $G$, a card, and call the multiset of all cards, the deck of $G$. The graph reconstruction problem is to determine the isomorphism type of $G$ by only looking at its deck.

Question: Is there a slick argument for reconstructing the number of Hamiltonian cycles of $G$ from its deck?

I have seen a paper of Tutte where he solves this using some machinery he uses to reconstruct several polynomial invariants of $G$. I don't have access to that paper right now, and I was wondering if a simpler solution exists when we restrict our attention just to counting Hamiltonian cycles.

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Bill Kocay found a more direct combinatorial method to reconstruct the number of hamiltonian cycles and some other spanning subgraphs. It is in his paper "Some new methods in reconstruction theory", Lecture Notes in Mathematics Volume 952, 1982, pp 89-114. You can get it here if you have sufficient Springer access.

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Thanks! That's exactly the kind of argument I was trying to find :) –  Gjergji Zaimi Aug 31 '13 at 0:45