Bipartite graphs whose edge algebras are complete intersections
  Let R be monomial sub-algebra of $k[x_1,...,x_N]$ generated by square free monomials of degree two. This paper addresses the following question: when is R a complete intersection?   For such a k-algebra we can associate a graph G whose vertices are $x_1,...,x_N$ and whose edges are $\{(x_i, x_j) | x_i x_j \in R \}$. Conversely, for any graph G with vertices $\{x_1,...,x_N\}$ we define the {\it edge algebra associated with G} as the sub-algebra of $k[x_1,...,x_N]$ generated by the monomials ${x_i x_j | (x_i,x_j) \text{is an edge of} G}$. We denote this monomial algebra by k[G].   This paper describes all bipartite graphs whose edge algebras are complete intersections. 
Mordechai Katzman