Simple estimation can be made if one considers selected bipartite graph Gi and then counts all the possible non-bipartite graphs that can be made from this graph Gi by adding inner edges in earch part of the graph Gi. So, if number of vortices in one part is x and number of vertices of another part is n-x, then the total number of new graphs for graph Gi is ((a b) being the binomial coefficient) 2^(x 2)*2(n-x 2)-1. After finding the minimum expression for x, the final estimation is that for each bipartite graph there are additional 2^n^2 non-bipartite graphs. So the part of bipartite graphs goes to zer when n goes to infinity.
From this result it is possible to get estimation of the total number of bipartite graphs on n vertices.
There are several article that gives exact computation of number of bipartite graphs, but no explicit formulas are given (found).