The binary actions of alternating groups

Given a conjugacy class $\mathcal{C}$ in a group $G$ we define a new graph, $\mathrm{\Gamma }(\mathcal{C})$, whose vertices are elements of $\mathcal{C}$; two vertices $g,h\in \mathcal{C}$ are connected in $\mathrm{\Gamma }(\mathcal{C})$ if $[g,h]=1$ and either $g{h}^{-1}$ or $h{g}^{-1}$ is in $\mathcal{C}$. We prove a lemma that relates the binary actions of the group $G$ to connectivity properties of $\mathrm{\Gamma }(\mathcal{C})$. This lemma allows us to give a complete classification of all binary actions when $G=A_n$, an alternating group on $n$ letters with $n\ge 5$.