Bounding graph diameters of solvable groups

We define a graph associated with a group G by letting nontrivial degrees be the vertices, and placing an edge between distinct degrees if they are not relatively prime. Using results in the literature, it is not difficult to show that when G is solvable and the graph is connected, its diameter is at most 4. Recent results suggest that this bound might be obtained. We show that in fact this diameter is at most 3, which is best possible.