Hamiltonian cycles on random lattices of arbitrary genus

TL;DR

This paper derives exact counts of Hamiltonian cycles on random lattices of any genus, revealing a linear relationship between the conformational exponent and genus, advancing understanding of graph enumeration on complex surfaces.

Contribution

It provides the first exact enumeration of Hamiltonian cycles on random graphs on surfaces of arbitrary genus, extending previous work limited to simpler topologies.

Findings

Exact number of Hamiltonian cycles on random torus graphs

Extension to arbitrary genus surfaces

Linear dependence of conformational exponent on genus

Abstract

A Hamiltonian cycle of a graph is a closed path that visits every vertex once and only once. It has been difficult to count the number of Hamiltonian cycles on regular lattices with periodic boundary conditions, e.g. lattices on a torus, due to the presence of winding modes. In this paper, the exact number of Hamiltonian cycles on a random trivalent fat graph drawn faithfully on a torus is obtained. This result is further extended to the case of random graphs drawn on surfaces of an arbitrary genus. The conformational exponent gamma is found to depend on the genus linearly.