I am interested in the properties of a class of bipartite graphs $G(X \cup Y, E)$ where all nodes in $X$ are 3-regular, all nodes in $Y$ are 2-regular, and $|X|=|2Y/3|$. First, Is this a well known class of graphs? Secondly,Is there an example of intractable computational problem restricted to this class of bipartite graphs?
Hard computational problem on special class of bipartite graphs
complexity theory;graph theory
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.