Orbital diameters of the symmetric and alternating groups

For a primitive group G acting on a finite set , we define the orbital diameter to be the maximum of the diameters of all orbital graphs of G. In this paper, we study the orbital diameters of symmetric and alternating groups. We give necessary numerical conditions for the orbital diameter to be bounded by some constant c and give precise descriptions of the actions for which the orbital diameter is bounded by 5. For each primitive action, we also either determine all orbital graphs of diameter 2 or give descriptions of infinite families of orbital graphs of diameter 2.