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arXiv · 2510.02473

Two Proofs of the Hamiltonian Cycle Identity

Hamilton Sawczuk·Edinah Gnang
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Abstract

The Hamiltonian cycle polynomial can be evaluated to count the number of Hamiltonian cycles in a graph. It can also be viewed as a list of all spanning cycles of length $n$. We adopt the latter perspective and present a pair of original proofs for the Hamiltonian cycle identity which relates the Hamiltonian cycle polynomial to the important determinant and permanent polynomials. The first proof is a more accessible combinatorial argument. The second proof relies on viewing polynomials as both linear algebraic and combinatorial objects whose monomials form lists of graphs. Finally, a similar identity is derived for the Hamiltonian path polynomial.

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Hamilton Sawczuk, Edinah Gnang. 2025-10-02. Two Proofs of the Hamiltonian Cycle Identity. https://arxiv.org/abs/2510.02473

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