The minimum number of Hamilton cycles in a hamiltonian threshold graph of a prescribed order

TL;DR

This paper determines the minimum number of Hamilton cycles in hamiltonian threshold graphs of a given order and characterizes the unique graph achieving this minimum, including its degree sequence.

Contribution

It establishes the exact minimum number of Hamilton cycles in hamiltonian threshold graphs and identifies the unique extremal graph with this property.

Findings

Minimum number of Hamilton cycles is $2^{loor{(n-3)/2}}$.

Unique extremal graph has a specific degree sequence.

This graph is also the smallest among all hamiltonian threshold graphs of order n.

Abstract

We prove that the minimum number of Hamilton cycles in a hamiltonian threshold graph of order $n$ is $2^{\lfloor (n-3)/2\rfloor}$ and this minimum number is attained uniquely by the graph with degree sequence $n-1,n-1,n-2,\ldots,\lceil n/2\rceil,\lceil n/2\rceil,\ldots,3,2$ of $n-2$ distinct degrees. This graph is also the unique graph of minimum size among all hamiltonian threshold graphs of order $n.$