Edge pancyclic Cayley graphs on symmetric group

Abstract

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