I am interested in the number of all graphs whose vertices are coloured in two colours (I would like to know the number of all improper colourings, not the number of bi-coloured graphs, see below). The problem can be solved in special cases (provided that the automorphism group is known) by applying Polya's theorem. Also, the number of bi-coloured graphs (where vertices of the same colour must not be adjacent) is known (Harary, 1958). I am particularly interested in connected graphs, but I would also be very happy about anything on trees. Thanks a lot! Ivo