  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.
