Make up a reasonable definition for the bipartite complement of a bipartite graph I am shooting from the hip here and seeing what sticks.
I tried this definition below which I am not sure works. If it doesn't, please, suggest more accurate definitions. 
The reason I need this is that I want to be able to replace the degrees in $(4,3,3,3,3,3,3,2,2)$ (which is a degree sequence of a bipartite graph)  with smaller numbers. 

Let $V$ be the set of vertices of a bipartite graph $G.$ Then $V$ is a union of two partite sets $X, Y$  and $\sum_{v_i \in X}deg(v_i) = \sum_{u_j \in Y}deg(u_j) = m.$
We can define graph $H = (A \cup B, F) = (\text {union of vertices, edges})$ to be bipartite complement of $G$ if $\sum_{d_i \in A}deg(d_i) = \sum_{e_j \in B}deg(e_j) = m.$
For example, suppose a bipartite(?) graph $T$ has partite sets of vertices represented by their degrees as $\{3, 4\}, \{2, 5\}.$ Then $T$'s bipartite complement has partite set of vertices represented by their degrees as $\{1, 6\}, \{1, 2, 4\}.$
 A: What you've written really doesn't make sense on a few levels. First of all, your definition of a bipartite complement of a graph is literally just another bipartite graph with the same number of edges. Additionally, in the example in the last paragraph that you give, I don't see have $\{3, 4\}$, $\{2, 5\}$ can be construed to be the degrees of all the vertices of a graph, no less a bipartite one (there aren't even five vertices, so how can a vertex have degree 5?). Also, with your first example, I don't exactly know what you mean by "able to replace the degrees in $(4,3,3,3,3,3,3,2,2)$...with smaller numbers".
Let me tell you what the ordinary definition of the bipartite complement of a bipartite graph is, and you can hopefully tell if this fits your needs. Let $G$ be a bipartite graph with parts $X$ and $Y$. Let $K(X, Y)$ be the complete bipartite graph with parts $X$ and $Y$. We define the bipartite complement of $G$ as $K(X, Y) - G$. That is, the bipartite complement of $G$ is the graph that has the same two parts as $G$ and has an edge between two vertices of different parts if and only if $G$ does not have an edge between those two vertices. This definition easily generalizes to $k$-partite graphs.
For example, let $G = C_6$, which is a bipartite graph as drawn below (with parts $X$ and $Y$). Then the bipartite complement of $G$, which I'll denote by $\overline{G}$, is composed of three disjoint edges:

