Two Proofs of the Hamiltonian Cycle Identity
Hamilton Sawczuk, Edinah Gnang
TL;DR
The paper investigates the Hamiltonian cycle polynomial, which lists Hamiltonian cycles as monomials, and presents two original proofs of the Hamiltonian cycle identity that expresses this polynomial in terms of det and per. The first proof is a direct combinatorial argument that analyzes edge monomials through functional graphs and cycle refinements to show that only Hamiltonian cycles contribute with coefficient 1. The second proof leverages Tutte's directed matrix tree theorem in a symbolic framework, using a determinant-sum lemma and multivariable derivatives to separate graphs by degree sequences and recover the same identity. Additionally, a parallel identity for the Hamiltonian path polynomial is derived. The work highlights deep connections between graph enumeration and classical algebraic objects like the determinant and permanent.
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.