Given a finite group $G$, the generating graph $\Gamma(G)$ 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 talk we investigate properties about the degrees of the vertices of $\Gamma(G)$ when $G$ is an alternating or a symmetric of degree $n$. In particular, we illustrate that $\Gamma(G)$ is Eulerian if and only if $n > 3$ and $n$ and $n-1$ are not equal to a prime number congruent to $3$ modulo $4$.  
We will also mention a couple of open problems. (Joint work with Andrea Lucchini.)