The directed genus of the de Bruijn graph

Define the directed genus  , Γ(G)Γ(G), of an Eulerian digraph GG to be the minimum value of pp for which GG has a 2-cell embedding in the orientable surface of genus pp so that every face of the embedding is bounded by a directed circuit in GG. The directed genus of the de Bruijn graph DnDn is shown to be Γ(Dn)=2n−1+1−12(n+2)∑d|n+2ϕ(d)2n+2d.