Edge pancyclic Cayley graphs on symmetric group

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.