Hard computational problem on special class of bipartite graphs
Given a 3-regular graph $G = \{V,E\}$ you can build a bipartite graph $G'$ with the required properties picking $X = V$ and $Y = E$ and for every edge $e_k = (u_i,u_j) \in E$ add edges $(u_i, e_k), (e_k, u_j)$. So I think that you can find some hard problems starting from hard problems on 3-regular graphs. 

For example SUBGRAPH ISOMORPHISM is NP-hard for your class of graphs.

The reduction is from Hamiltonian cycle on 3-regular graphs: given a 3-regular graph $G$, build the corresponding $G' = \{X \cup Y, E'\}$ and check for a subgraph $H'$ which is a closed simple cycle of length $2|V|$. $G'$ has a subgraph isomorphic to $H'$ if and only if $G$ has an Hamiltonian cycle.