Upper Bounds for the Number of Hamiltonian Cycles

TL;DR

This paper establishes tighter upper bounds for the number of Hamiltonian cycles in symmetric directed graphs and extends these bounds to undirected graphs via a graph transformation method.

Contribution

It introduces a new, tighter upper bound for Hamiltonian cycles in symmetric diagraphs and a transformation technique to extend this bound to undirected graphs.

Findings

Tighter upper bound for symmetric diagraphs.

Transformation method for undirected graphs.

Upper bound for undirected graphs derived.

Abstract

An upper bound for the number of Hamiltonian cycles of symmetric diagraphs is established first in this paper, which is tighter than the famous Minc's bound and the Br$\acute{e}$gman's bound. A transformation on graphs is proposed, so that counting the number of Hamiltonian cycles of an undirected graph can be done by counting the number of Hamiltonian cycles of its corresponding symmetric directed graph. In this way, an upper bound for the number of Hamiltonian cycles of undirected graphs is also obtained.