Edge pancyclic derangement graphs

TL;DR

This paper proves that derangement graphs, where vertices are permutations differing in all positions, are edge pancyclic for all n ≥ 4, extending known Hamiltonian properties.

Contribution

It establishes that derangement graphs are edge pancyclic for all n ≥ 4, a new property beyond their known Hamiltonian characteristics.

Findings

Derangement graphs are Hamiltonian and Hamilton-connected.

For n ≥ 4, derangement graphs are edge pancyclic.

The result extends understanding of the cycle structure in derangement graphs.

Abstract

We consider the derangement graph in which the vertices are permutations of $\{ 1,\ldots, n\}$. Two vertices are joined by an edge if the corresponding permutations differ in every position. The derangement graph is known to be Hamiltonian and Hamilton-connected. In this note, we show that the derangement graph is edge pancyclic if $n\ge 4$.