Counting Hamiltonian cycles on planar random lattices

TL;DR

This paper studies Hamiltonian cycles on planar random lattices, deriving their generating function, analyzing its singularity, and exploring connections to two-dimensional quantum gravity.

Contribution

It introduces a generating function for Hamiltonian cycles on planar lattices and investigates its singularity, linking combinatorics with quantum gravity.

Findings

Derived the generating function for Hamiltonian cycles

Analyzed the singularity structure of the generating function

Discussed implications for two-dimensional quantum gravity

Abstract

A Hamiltonian cycle of a graph is a closed path which visits each of the vertices once and only once. In this article, Hamiltonian cycles on planar random lattices are considered. The generating function for the number of Hamiltonian cycles is obtained and its singularity is studied. Relation to two-dimensional quantum gravity is discussed.