Hamilton Powers of Eulerian Digraphs

TL;DR

This paper proves that a certain power of connected Eulerian digraphs is Hamiltonian, establishing a threshold and providing examples where lower powers are not Hamiltonian, advancing understanding of Hamiltonian properties in digraphs.

Contribution

It introduces a specific power threshold for Hamiltonicity in connected Eulerian digraphs and constructs examples showing the threshold's sharpness.

Findings

The $ ceil rac{1}{2} oot{n} \log_2^2 n ceil$-th power of a connected $n$-vertex Eulerian digraph is Hamiltonian.

An infinite family of digraphs exists where the $loor{rac{ oot{n}}{2}}$-th power is not Hamiltonian.

Abstract

In this note, we prove that the $\lceil \tfrac{1}{2} \sqrt{n} \log_2^2 n \rceil^{th}$ power of a connected $n$-vertex Eulerian digraph is Hamiltonian, and provide an infinite family of digraphs for which the $\lfloor \sqrt{n}/2 \rfloor^{th}$ power is not.