  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 paper we
investigate properties about the degrees of the vertices of $\Gamma(G)$ when
$G$ is an alternating group or a symmetric group. In particular, we determine
the vertices of $\Gamma(G)$ having even degree and show that $\Gamma(G)$ is
Eulerian if and only if $n$ and $n-1$ are not equal to a prime number congruent
to 3 modulo 4.
