  In this paper we are concerned with the classification of the finite groups
admitting a bipartite DRR and a bipartite GRR.
  First, we find a natural obstruction in a finite group for not admitting a
bipartite GRR. Then we give a complete classification of the finite groups
satisfying this natural obstruction and hence not admitting a bipartite GRR.
Based on these results and on some extensive computer computations, we state a
conjecture aiming to give a complete classification of the finite groups
admitting a bipartite GRR.
  Next, we prove the existence of bipartite DRRs for most of the finite groups
not admitting a bipartite GRR found in this paper. Actually, we prove a much
stronger result: we give an asymptotic enumeration of the bipartite DRRs over
these groups. Again, based on these results and on some extensive computer
computations, we state a conjecture aiming to give a complete classification of
the finite groups admitting a bipartite DRR.
