Compatible eulerian circuits in

Let Kn∗∗ be the complete symmetric digraph with a loop at each vertex. We say that two eulerian circuits of Kn∗∗ are compatible if no pair of arcs of the digraph are consecutive in both eulerian circuits. We prove that there exist φ(n) pairwise compatible eulerian circuits in Kn∗∗, where φ denotes the Euler function, and we give an effective construction of these circuits.