A particular kind of 2-cell imbedding for a class of edge-coloured graphs into surfaces with boundary is introduced and studied. This allows to define, as in [Geom. Dedicata 11 (1981), 397-414], where the closed case was treated, a pair of invariants -the "regular genus" and the "hole-number" - for every n-manifold with boundary. These invariants are proved to coincide with the classical ones in dimension two, and to be strictly related with a Heegaard-like handlebody decomposition in dimension three. A characterization of the n-disk D^n, as the unique n-manifold with regular genus zero and hole-number one, concludes the work.