Field theoretic approach to the counting problem of Hamiltonian cycles of graphs

TL;DR

This paper introduces a field theoretic method to estimate the number of Hamiltonian cycles in graphs, providing a novel analytical approach validated on 2D square lattices with different boundary conditions.

Contribution

It develops a new field theoretic representation for counting Hamiltonian cycles and demonstrates its effectiveness through quadratic approximation and empirical validation.

Findings

Quadratic fluctuations yield meaningful estimates of Hamiltonian cycle counts.

The method accurately predicts counts for 2D square lattices with various boundary conditions.

First successful application of quadratic approximation in this context.

Abstract

A Hamiltonian cycle of a graph is a closed path that visits each site once and only once. I study a field theoretic representation for the number of Hamiltonian cycles for arbitrary graphs. By integrating out quadratic fluctuations around the saddle point, one obtains an estimate for the number which reflects characteristics of graphs well. The accuracy of the estimate is verified by applying it to 2d square lattices with various boundary conditions. This is the first example of extracting meaningful information from the quadratic approximation to the field theory representation.