Regular embeddings of coloured digraphs

A particular kind of 2-cell embeddings, called regular, for arc-coloured digraphs is introduced, and a method for constructing both orientable and non-orientable regular embeddings is presented. Furthermore, by using combinatorial concepts and the Euler-Poincaré formula, we derive upper bounds for both the orientable and non-orientable genera of such arc-coloured digraphs.Work performed within the project Geometria delle Varietà Differenziabili of the MPI of Italy.