Edge pancyclic Cayley graphs on symmetric group

TL;DR

This paper proves that derangement graphs on the symmetric group are edge pancyclic for all n ≥ 4 and extends this property to broader classes of Cayley graphs on the same group.

Contribution

It establishes the edge pancyclicity of derangement graphs on symmetric groups and generalizes the result to other Cayley graphs on the same group.

Findings

Derangement graphs are edge pancyclic for n ≥ 4.

The edge pancyclic property extends to broader Cayley graphs on symmetric groups.

Derangement graphs are Hamiltonian and Hamilton-connected.

Abstract

We study the derangement graph $\Gamma_n$ whose vertex set consists of all permutations of $\{1,\ldots,n\}$, where two vertices are adjacent if and only if their corresponding permutations differ at every position. It is well-known that $\Gamma_n$ is a Cayley graph, Hamiltonian and Hamilton-connected. In this paper, we prove that for $n \geq 4$, the derangement graph $\Gamma_n$ is edge pancyclic. Moreover, we extend this result to two broader classes of Cayley graphs defined on symmetric group.