The directed genus of the de Bruijn graph

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