# Constructing Arbitrarily Large Graphs with a Specified Number of Hamiltonian Cycles

## TL;DR

The paper presents a constructive method to generate large directed graphs with a specified number of Hamiltonian cycles, useful for testing algorithms on complex instances of the Hamiltonian cycle problem.

## Contribution

It introduces a novel construction called broken crown graphs that can produce graphs with any number of Hamiltonian cycles for given parameters.

## Key findings

- Graphs can have arbitrarily many Hamiltonian cycles.
- Constructed graphs remain non-trivial for small k.
- Method applies to large directed graphs with controlled cycle counts.

## Abstract

A constructive method is provided that outputs a directed graph which is named a broken crown graph, containing $5n-9$ vertices and $k$ Hamiltonian cycles for any choice of integers $n \geq k \geq 4$. The construction is not designed to be minimal in any sense, but rather to ensure that the graphs produced remain non-trivial instances of the Hamiltonian cycle problem even when $k$ is chosen to be much smaller than $n$.