Alternating and symmetric groups with Eulerian generating graph

Abstract

Given a finite group $G$, the generating graph $Γ(G)$ of $G$ has as vertices the (nontrivial) elements of $G$ and two vertices are adjacent if and only if they are distinct and generate $G$ as group elements. In this paper we investigate properties about the degrees of the vertices of $Γ(G)$ when $G$ is an alternating group or a symmetric group. In particular, we determine the vertices of $Γ(G)$ having even degree and show that $Γ(G)$ is Eulerian if and only if $n$ and $n-1$ are not equal to a prime number congruent to 3 modulo 4.